Class 6 Mathematics Chapter 1 Patterns in Mathematics

Key Concept: Patterns in Nature, Mathematical Applications

a) Asymptotic convergence
[Solution Description]
The Fibonacci sequence has the property that the ratio of consecutive terms approaches the golden ratio $\phi$. This is derived from solving the characteristic equation of the recurrence relation, which gives $\phi$ as one of its solutions. The golden ratio is an irrational number approximately equal to 1.61803, often observed in natural patterns.

Key Concept: Mathematics as Art and Science

c) It combines creative pattern-finding with logical reasoning
[Solution Description]
Mathematics is an art because it involves creativity in discovering patterns, and a science because it seeks logical explanations for those patterns. This duality is fundamental to mathematical practice.

Key Concept: Applications of Mathematical Patterns

b) By enabling efficient data storage and processing in computers
[Solution Description]
Mathematical patterns form the basis of many technologies. For example, understanding patterns in binary code allows computers to process information. Similarly, patterns in algorithms enable efficient data processing, and patterns in electrical signals help in communication technologies. Thus, mathematical patterns are fundamental to the functioning of computers and mobile phones.

Key Concept: Patterns in Mathematics, Applications of Patterns

c) Assertion is true, but Reason is false.
[Solution Description]
The Fibonacci sequence is indeed observed in various biological systems such as the arrangement of leaves, petals, and seeds due to its efficiency in packing and growth. The assertion correctly states this observation. However, the reason provided incorrectly links the Fibonacci sequence solely to minimizing energy consumption and maximizing sunlight exposure. While these are factors in plant growth, they are not the primary explanation for the appearance of the Fibonacci sequence in nature. The sequence arises from the mathematical properties of recursive addition rather than energy optimization. Thus, the Assertion is true but the Reason is false.

Key Concept: Patterns in Architecture

b) A ratio of approximately 1.618:1
[Solution Description]
The Golden Ratio, approximately 1.618, is a special number found by dividing a line into two parts so that the longer part divided by the smaller part equals the whole length divided by the longer part. It appears in many architectural designs for aesthetic harmony.

Key Concept: Number Sequences and Shapes

b) Heptagon
[Solution Description]
Regular polygons are named based on the number of sides they have. A 3-sided polygon is a triangle, 4-sided is quadrilateral (square), 5-sided is pentagon, 6-sided is hexagon, and 7-sided is heptagon.

Key Concept: Pattern Recognition, Real-World Application

c) 36
[Solution Description]
Start with the first number: 3.
To get the second number, add 4: $3 + 4 = 7$, then multiply by 2: $7 \times 2 = 14$.
To get the third number, add 4 to the second number: $14 + 4 = 18$, then multiply by 2: $18 \times 2 = 36$.

Key Concept: Mathematics as Art and Science

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The syllabus states that mathematics is both an art and a science due to its creative aspect in discovering patterns and explaining them. The assertion correctly identifies mathematics as an art because of the creativity involved in pattern discovery, which aligns with the syllabus. The reason further explains why this creative process makes mathematics an art, as it goes beyond simple calculations. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Mathematical Patterns in Infrastructure

c) By calculating load distributions and stress points
[Solution Description]
Structural engineers use mathematical patterns to calculate load distributions and stress points. This ensures that bridges and houses can withstand forces like wind and weight without collapsing.

Key Concept: Mathematical Patterns in Health

c) By understanding mutations and disease mechanisms
[Solution Description]
Identifying patterns in genetic sequences helps scientists understand mutations and disease mechanisms. This leads to better diagnosis and targeted treatments for various illnesses.

Key Concept: Patterns, Applications

c) 1600
[Solution Description]
The pattern described is exponential growth. The formula is:
$Final Population = Initial Population \times 2^{(Time/Doubling Time)}$
Here, Initial Population = 100, Doubling Time = 3 hours, Time = 12 hours.
Step 1: Calculate exponent: $12/3 = 4$.
Step 2: Apply formula: $100 \times 2^4 = 100 \times 16 = 1600$.

Key Concept: Mathematical Patterns in Daily Life

b) Cooking with measured ingredients
[Solution Description]
Many daily activities involve recognizing and using mathematical patterns without explicit awareness. Cooking requires measuring ingredients in ratios or proportions, scheduling involves time management (a form of arithmetic), shopping uses budgeting and discounts (percentage calculations), and playing games often includes strategic counting or spatial reasoning. Among the options, cooking accurately with measured ingredients is the most direct application of mathematical patterns.

Key Concept: Economic Indicator

b) Rise in general price levels
[Solution Description]
Inflation measures the rate at which prices of goods and services rise over time, reducing purchasing power.

Key Concept: Infrastructure Investment

c) 3.0
[Solution Description]
The multiplier effect is calculated as the change in GDP divided by the initial investment. Here, GDP increases by \$1.5 billion (\$1500 million) due to an initial investment of \$500 million.
Multiplier = $\frac{\text{Change in GDP}}{\text{Initial Investment}} = \frac{1500}{500} = 3$.
Thus, the multiplier effect is 3.

Key Concept: Medical Equipment

c) CT scan
[Solution Description]
Computed Tomography (CT) scans use X-rays to produce detailed cross-sectional images of the body’s internal structures.

Key Concept: General

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that biomedical engineering combines engineering and medical sciences, which is true. The reason explains its focus on healthcare improvement through technology, which is also true. The reason correctly explains the assertion since biomedical engineering aims to bridge engineering and medicine for healthcare advancements.

Key Concept: Hexagonal numbers, Cumulative sums

c) 64
[Solution Description]
From the syllabus, the sequence of hexagonal numbers is: 1, 7, 19, 37,…
We need to find $1 + 7 + 19 + 37$.
Step 1: $1 + 7 = 8$
Step 2: $8 + 19 = 27$
Step 3: $27 + 37 = 64$.

Key Concept: Powers of 3, Virahānka numbers

d) 5
[Solution Description]
The given alternating sequence has powers of 3 ($3^0, 3^1, 3^2,…$) at odd positions and Virahānka numbers (Fibonacci: 1,1,2,3,5,8,…) at even positions.
Positions:
1st: $3^0=1$
2nd: Virahānka 1st=1
3rd: $3^1=3$
4th: Virahānka 2nd=1
5th: $3^2=9$
6th: Virahānka 3rd=2
7th: $3^3=27$
8th: Virahānka 4th=3
9th: $3^4=81$
10th: Virahānka 5th=5.

Key Concept: Square Numbers

c) 25
[Solution Description]
The given sequence represents square numbers. Each term is the square of the next integer. The sequence corresponds to $1^2, 2^2, 3^2, 4^2,$ so the next term is $5^2 = 25$.
Calculation: $5 \times 5 = 25$.

Key Concept: Triangular numbers, Square numbers

b) 11
[Solution Description]
The sum of the first $n$ odd numbers equals $n^2$. Here $n = 7$, so sum is $7^2 = 49$.
Triangular numbers are given by $T_k = \frac{k(k+1)}{2}$. We need to find $k$ and $m$ such that $49 = T_k + m^2$. Checking possible values:
For $k = 6$, $T_6 = 21$. Then $m^2 = 49 – 21 = 28$, but 28 isn’t a perfect square.
For $k = 8$, $T_8 = 36$. Then $m^2 = 49 – 36 = 13$, not a perfect square.
For $k = 7$, $T_7 = 28$. Then $m^2 = 49 – 28 = 21$, not a perfect square.
For $k = 9$, $T_9 = 45$. Then $m^2 = 49 – 45 = 4$, so $m = 2$.
Thus $k + m = 9 + 2 = 11$.

Key Concept: Triangular Numbers

c) 15
[Solution Description]
Triangular numbers are generated by adding consecutive natural numbers. The sequence starts with 1 (term 1), then 1+2=3 (term 2), 1+2+3=6 (term 3), 1+2+3+4=10 (term 4). For the 5th term, we add the next natural number to the previous sum: $10 + 5 = 15$.

Key Concept: Virahānka numbers

b) 34
[Solution Description]
The Virahānka sequence is formed by adding the two previous terms to get the next term.
Given: $1, 2, 3, 5, 8, 13, 21, \dots$
To find the next term: $13 + 21 = 34$
So, the next term is $34$.

Key Concept: Counting Numbers Application

c) 5
[Solution Description]
To find the total number of apples, add the initial count (3) to the new apples (2). So, 3 + 2 = 5. Counting numbers are used here for simple addition.

Key Concept: Triangular numbers

c) 15
[Solution Description]
Triangular numbers are formed by adding consecutive counting numbers.
For the 5th triangular number:
Step 1: Start with 1 dot for first row
Step 2: Add 2 dots for second row making total 3 dots
Step 3: Add 3 dots for third row making total 6 dots
Step 4: Add 4 dots for fourth row making total 10 dots
Step 5: Add 5 dots for fifth row making total 15 dots
So the correct answer is $15$.

Key Concept: Visual representation, Odd number properties

c) 36
[Solution Description]
The number of dots in the $n^{th}$ layer of the grid is the $n^{th}$ odd number, which is given by $2n – 1$. The total dots in $6$ layers is the sum of the first $6$ odd numbers: $1 + 3 + 5 + 7 + 9 + 11 = 36$. Alternatively, using the property that the sum of the first $n$ odd numbers is $n^2$, the answer is $6^2 = 36$.

Key Concept: Pattern in Sum of Odd Numbers

b) 121
[Solution Description]
We know that the sum of the first $n$ odd numbers is equal to $n^2$.
Given that the sum of the first 10 odd numbers is $10^2 = 100$, the sum of the first 11 odd numbers will be $11^2 = 121$.
Alternatively, since the 11th odd number is $21$ (since the sequence is 1, 3, 5, …, 19, 21), we can add it to the previous sum: $100 + 21 = 121$.
Therefore, the sum of the first 11 odd numbers is $121$.

Key Concept: Basic Even Number Addition

c) 6
[Solution Description]
The first two even numbers are 2 and 4. The sum is calculated as:
$2 + 4 = 6$
Thus, the correct answer is 6.

Key Concept: Even Number Sequence

b) 18, 20
[Solution Description]
The given sequence is of even numbers increasing by 2 each time. The last given number is 16. To find the next number, add 2 to 16: $16 + 2 = 18$. Then add 2 again to get the following number: $18 + 2 = 20$. So, the next two numbers are 18 and 20.

Key Concept: Triangular numbers, Square numbers

b) When $\sqrt{8n^2 + 1}$ is an integer
[Solution Description]
We look for solutions where $T_n = k^2$ for some integer $k$. Substituting $T_n$:
$\frac{n(n+1)}{2} = k^2 \implies n^2 + n – 2k^2 = 0.$
Solving this Diophantine equation gives Pell-like solutions.
Known examples include $T_1 = 1 = 1^2$ and $T_8 = 36 = 6^2$.
The condition occurs when $\sqrt{8k^2 + 1}$ is odd.
Thus, the correct statement involves finding $n$ such that $\sqrt{8k^2 + 1}$ is an integer.

Key Concept: Triangular numbers, Hexagonal numbers

a) $H_n = n^3 – (n-1)^3$
[Solution Description]
To find the relationship between the $n^{th}$ hexagonal number ($H_n$) and the $n^{th}$ triangular number ($T_n$), we observe the pattern:
– First hexagonal number ($H_1$): 1 = $3 \times 1 – 2 \times 1 = 1$
– Second hexagonal number ($H_2$): 7 = $3 \times 4 – 2 \times 2.5 = 7$ (Not matching). Let’s consider another approach.
The general formula for the $n^{th}$ hexagonal number is $H_n = 3n^2 – 3n + 1$.
Comparing this with triangular numbers, we have:
$H_n = 3T_{2n-1} – 2T_{n-1}$ is incorrect. Instead, we can verify:
For $n = 1$: $H_1 = 1$, $T_1 = 1$ → $H_1 = 6T_1 – 5 \times 0 = 1$ (Not consistent).
Correct relationship: Each hexagonal number is equal to $3n^2 – 3n + 1$, which is not directly expressible in terms of $T_n$. However, we can rewrite:
$H_n = n^3 – (n-1)^3$, e.g., for $n=2$: $8 – 1 = 7 = H_2$.
Hence, the correct relationship is $H_n = n^3 – (n-1)^3$.

Key Concept: Square numbers, Sum of odd numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sum of the first $n$ odd numbers is a perfect square. Let’s verify this with examples:
For $n = 1$: $1 = 1^2$

For $n = 2$: $1 + 3 = 4 = 2^2$

For $n = 3$: $1 + 3 + 5 = 9 = 3^2$

This pattern holds true for any natural number $n$, as per the syllabus. The reason explains that square numbers can be represented as the sum of consecutive odd numbers starting from 1. Since the assertion and reason are both true and the reason correctly explains the assertion, the correct answer is option (a).

Key Concept: Square Numbers, Pattern Recognition

b) Differences increase by 2 each time
[Solution Description]
The sequence of squares of consecutive integers is: $1, 4, 9, 16, 25, \ldots$.
The differences between consecutive terms are:
$4 – 1 = 3$,
$9 – 4 = 5$,
$16 – 9 = 7$,
$25 – 16 = 9$,
$\ldots$
These differences form the sequence of odd numbers starting from 3.

Key Concept: Virahānka numbers sequence

c) 34
[Solution Description]
The Virahānka (Fibonacci-type) sequence is formed by adding the two previous numbers to get the next number. Here, the last two numbers are $13$ and $21$. So, the next number is $13 + 21 = 34$.

Key Concept: Virahānka sequence rule

b) $2a + 3b$
[Solution Description]
Given the first three numbers: $a, b, a + b$.
The fourth number = second + third number = $b + (a + b) = a + 2b$.
The fifth number = third + fourth number = $(a + b) + (a + 2b) = 2a + 3b$.

Key Concept: Square Numbers

b) 85
[Solution Description]
The sequence of square numbers is given by $n^2$ where $n$ is a positive integer. The 6th square number is $6^2 = 36$, and the 7th square number is $7^2 = 49$. The sum of these two numbers is $36 + 49 = 85$.

Key Concept: Koch Snowflake, Powers of 4

c) 192
[Solution Description]
The sequence of line segments in the Koch Snowflake is given by multiplying by $4$ at each step. Starting with $3$ segments:
\begin{align*}
1^{st} \text{ iteration}: 3
2^{nd} \text{ iteration}: 3 \times 4 = 12
3^{rd} \text{ iteration}: 12 \times 4 = 48
4^{th} \text{ iteration}: 48 \times 4 = 192
\end{align*}
Thus, the number of line segments in the $4^{th}$ iteration is $192$.

Key Concept: Triangular numbers, visual representation

c) 5
[Solution Description]
For triangular numbers, the pattern grows by adding one more dot than the previous row. For the fifth triangular number (15), the rows are as follows: Row 1 has 1 dot, Row 2 has 2 dots, Row 3 has 3 dots, Row 4 has 4 dots, and Row 5 has 5 dots. Thus, the fifth row will have 5 dots.

Key Concept: Visualising Square Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion claims that 16 is a square number and can be represented as a $4 \times 4$ grid of dots. This is true because $4^2 = 16$, and square numbers are defined as numbers that can form a perfect square with dots (e.g., $1, 4, 9, 16, \dots$). The reason states that square numbers are formed by arranging dots in perfect squares, which is also true and directly explains why the assertion is correct.

Key Concept: Patterns in Numbers, Visualizing Number Sequences

c) 15
[Solution Description]
To find the 5th term:
1. Start with the first term: $a_1 = 1$.
2. Second term: $a_2 = a_1 + 2 = 1 + 2 = 3$.
3. Third term: $a_3 = a_2 + 3 = 3 + 3 = 6$.
4. Fourth term: $a_4 = a_3 + 4 = 6 + 4 = 10$.
5. Fifth term: $a_5 = a_4 + 5 = 10 + 5 = 15$.
The 5th term is 15.

Key Concept: Pattern Recognition in Nature

a) Movement of tides
[Solution Description]
The question asks about a natural phenomenon demonstrating a predictable and consistent mathematical pattern.
– The movement of tides is driven by gravitational forces, primarily from the moon and sun, which follow Newton’s laws of motion and gravity, making them predictable.
– Bird migration, while seasonal, varies based on environmental factors and is not strictly mathematical.
– Cloud formation depends on chaotic weather patterns and lacks strict predictability.
– Volcanic eruptions are geological events with no fixed timing or pattern.
Thus, tides are the most mathematically predictable among the options.

Key Concept: Square Numbers – Sum of Odd Numbers

c) 9
[Solution Description]
The sum of the first 3 odd numbers (1, 3, 5) is $1 + 3 + 5 = 9$. This is also equal to the square of 3, i.e., $3^2 = 9$.

Key Concept: Pattern of Counting Number Sums

b) 25
[Solution Description]
The given sum follows the pattern where the sum of counting numbers up and then down forms a square number. Calculating step by step:
$1 + 2 + 3 + 4 + 5 = 15$ and $4 + 3 + 2 + 1 = 10$. Total sum is $15 + 10 = 25$. Thus, the square number obtained is $25$.

Key Concept: Triangular Numbers, Sum of Consecutive Triangular Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The nth triangular number is given by the formula $T_n = \frac{n(n+1)}{2}$. Let’s verify the relationship between consecutive triangular numbers:
Step 1: Compute $T_n + T_{n+1}$.
$T_n + T_{n+1} = \frac{n(n+1)}{2} + \frac{(n+1)(n+2)}{2}$
Step 2: Factor out common terms.
$T_n + T_{n+1} = \frac{(n+1)(n + n + 2)}{2} = \frac{(n+1)(2n + 2)}{2}$
Step 3: Simplify further.
$T_n + T_{n+1} = \frac{2(n+1)(n + 1)}{2} = (n+1)^2$
Thus, the sum of two consecutive triangular numbers equals a perfect square $(n+1)^2$. Therefore, the Assertion and Reason are both true, and Reason correctly explains the Assertion.

Key Concept: Sum of Consecutive Triangular Numbers

c) A square number.
[Solution Description]
Adding consecutive triangular numbers gives square numbers:
– $1 + 3 = 4$ (which is $2^2$),
– $3 + 6 = 9$ (which is $3^2$),
– $6 + 10 = 16$ (which is $4^2$),
This pattern continues for higher triangular numbers as well.

Key Concept: Shape sequences, multi-concept challenge

b) 4
[Solution Description]
Each step adds a layer of cubes around the previous structure. After four steps, the number of cubes used in each step forms the sequence $1, 8, 27, 64$ (the first four cube numbers). The largest cube formed has a side length equal to the cube root of the total number of cubes: $\sqrt[3]{1 + 8 + 27 + 64} = \sqrt[3]{100}$. However, the problem asks for the side length ratio of the largest to smallest cube. The side lengths in each step correspond to the cube roots of the number of new cubes added: $\sqrt[3]{1}$, $\sqrt[3]{8}=2$, $\sqrt[3]{27}=3$, $\sqrt[3]{64}=4$. Thus, the ratio of the largest to smallest side after four steps is $\frac{4}{1} = 4$.

Key Concept: Visualising number sequences, cube numbers

b) $(1 + 2 + 3)^2$
[Solution Description]
The first three cube numbers are $1^3 = 1$, $2^3 = 8$, and $3^3 = 27$. Their sum is $1 + 8 + 27 = 36$. This sum can also be represented as $\left(1 + 2 + 3\right)^2 = 6^2 = 36$. So, the sum equals the square of the sum of their base numbers.

Key Concept: Square Number Pattern

c) 25
[Solution Description]
This pattern generates square numbers. Each sum corresponds to the square of the largest number in the middle of the sequence. For example: $1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4^2$. The next pattern would be $1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1$, which equals $25 = 5^2$.

Key Concept: Generalizing Sum of Odd Numbers, Square Numbers

c) 1875
[Solution Description]
The sum of the first $m$ odd numbers is $m^2 = 625$. Thus, $m = 25$. The next $m$ odd numbers start from $(2m+1)$, i.e., the 26th odd number (which is $51$) to the 50th odd number (which is $99$). The sum of these $m$ terms can be calculated as the difference between the sum of the first $2m$ odd numbers and the sum of the first $m$ odd numbers:
$\text{Sum of first } 2m \text{ odd numbers} = (2m)^2 = 2500$
$\text{Sum of next } m \text{ odd numbers} = 2500 – 625 = 1875$
Alternatively, the sum of the next $m$ odd numbers after the first $m$ odd numbers can be generalized as $3m^2$:
$3 \times 25^2 = 3 \times 625 = 1875$

Key Concept: Sum of odd numbers, square numbers, algebraic expression

c) $S_k + (2k + 1)$
[Solution Description]
The sum of the first $k$ odd numbers is $S_k = k^2$. The $(k+1)$-th odd number is $2k + 1$. Therefore, the sum of the first $(k+1)$ odd numbers is:
$S_{k+1} = S_k + (2k + 1) = k^2 + 2k + 1 = (k + 1)^2$
So, $S_{k+1}$ is obtained by adding the $(k+1)$-th odd number to $S_k$.

Key Concept: Sum of odd numbers, square numbers, real-world application

c) 41
[Solution Description]
The problem describes a scenario where the sum of the first $m$ odd numbers equals 441. As the sum of the first $m$ odd numbers is $m^2$, we have:
$m^2 = 441$
Solving for $m$:
$m = \sqrt{441} = 21$
The $m$-th odd number corresponds to the number of tiles added in the $m$-th layer. The $m$-th odd number is given by $2m – 1$. Substituting $m = 21$:
$2(21) – 1 = 42 – 1 = 41$

Key Concept: Symmetrical Addition, Generalization

b) $S(n) = n^2$
[Solution Description]
From the pattern observed in symmetrical addition, we see that $1 + 2 + 1 = 4 = 2^2$, $1 + 2 + 3 + 2 + 1 = 9 = 3^2$, and so on. Thus, the general formula for $S(n)$ is $n^2$.

Key Concept: Symmetrical addition, Square numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the symmetrical sum of counting numbers up to $n$ and back down to 1 equals $n^2$. This is true because the sum can be split into two parts: the ascending part ($1 + 2 + \ldots + n$) and the descending part ($(n-1) + (n-2) + \ldots + 1$). The sum of the ascending part is $\frac{n(n+1)}{2}$, and the sum of the descending part is $\frac{(n-1)n}{2}$. Adding these two gives $\frac{n(n+1)}{2} + \frac{(n-1)n}{2} = \frac{n^2 + n + n^2 – n}{2} = n^2$, which matches the assertion. The reason provides a visual explanation: arranging the numbers in layers forms a square grid of size $n \times n$, which directly represents $n^2$. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Generalization, Triangular numbers, Square numbers

b) $n+1$
[Solution Description]
The $n^{th}$ triangular number is $T_n = \frac{n(n+1)}{2}$, and the $(n+1)^{th}$ triangular number is $T_{n+1} = \frac{(n+1)(n+2)}{2}$. Their sum is:
$T_n + T_{n+1} = \frac{n(n+1)}{2} + \frac{(n+1)(n+2)}{2} = \frac{(n+1)(2n+2)}{2} = (n+1)^2$
Thus, the sum is always $(n+1)^2$, meaning the side length of the square is $(n+1)$.

Key Concept: Identifying Triangular Pairs

c) 21 and 28
[Solution Description]
To identify which pair of consecutive triangular numbers adds up to $49$:
1. From the syllabus, we know that adding two consecutive triangular numbers yields a square number.
2. First, recognize that $49$ is $7^2$. So, the sum must be from the 6th and 7th triangular numbers.
3. The 6th triangular number is $21$ ($1+2+3+4+5+6=21$).
4. The 7th triangular number is $28$ ($1+2+3+4+5+6+7=28$).
5. Verify: $21 + 28 = 49$.

Key Concept: Generalizing the pattern, Powers of 3

b) $\frac{3^n + 1}{2}$
[Solution Description]
The sum of the first $n$ powers of 3 is given by $S = 1 + 3 + 9 + \dots + 3^{n-1}$. This is a geometric series with sum $S = \frac{3^n – 1}{2}$. Adding 1 to this sum gives $S + 1 = \frac{3^n – 1}{2} + 1 = \frac{3^n + 1}{2}$. Therefore, the final number is $\frac{3^n + 1}{2}$.

Key Concept: Pattern Recognition

b) $2^5$
[Solution Description]
The first 5 powers of 2 are $1$, $2$, $4$, $8$, and $16$. Their sum is:
$1 + 2 + 4 + 8 + 16 = 31$
Adding 1 to this sum gives:
$31 + 1 = 32$
Since $32$ is $2^5$, the correct answer is $2^5$.

Key Concept: Generating sequences

b) $7, 19, 37, 61, \dots$
[Solution Description]
For the given triangular numbers:
– $1 \times 6 + 1 = 7$
– $3 \times 6 + 1 = 19$
– $6 \times 6 + 1 = 37$
– $10 \times 6 + 1 = 61$
Thus, the correct sequence is $7, 19, 37, 61, \dots$.

Key Concept: Triangular numbers, Hexagonal numbers

d) Assertion is false, but Reason is true.
[Solution Description]
To verify this, let us take the $n$-th triangular number $T_n = \frac{n(n+1)}{2}$. Multiplying this by 6 and adding 1 gives:
$6T_n + 1 = 6 \left( \frac{n(n+1)}{2} \right) + 1 = 3n(n+1) + 1 = 3n^2 + 3n + 1.$
Now, the formula for the $n$-th hexagonal number is given as $H_n = 3n^2 – 3n + 1$. Comparing both expressions:
$3n^2 + 3n + 1 \neq 3n^2 – 3n + 1.$
The Assertion claims that multiplying triangular numbers by 6 and adding 1 gives hexagonal numbers, but our calculation shows a discrepancy in the formulas. Hence, the Assertion is false. However, the Reason correctly states the formula for the $n$-th hexagonal number, so it is true.

Key Concept: Hexagonal Numbers and Cubes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion and reason step by step. The first hexagonal number is $1$, and $1^3 = 1$. The sum of the first two hexagonal numbers is $1 + 7 = 8$, and $2^3 = 8$. Similarly, the sum of the first three hexagonal numbers is $1 + 7 + 19 = 27$, and $3^3 = 27$. This pattern holds for higher values of $n$ as well, confirming that the sum of the first $n$ hexagonal numbers equals $n^3$ (assertion is true). The reason explains this by stating that each hexagonal number represents a new layer added to a cube, resulting in the next cube number, which correctly describes the observed pattern (reason is true and correctly explains the assertion).

Key Concept: Hexagonal numbers formula and proof

c) 64
[Solution Description]
Using the formula $H_n = 3n^2 – 3n + 1$, we calculate each term for $n = 1$ to $4$:
For $n = 1$: $H_1 = 3(1)^2 – 3(1) + 1 = 1$.
For $n = 2$: $H_2 = 3(2)^2 – 3(2) + 1 = 7$.
For $n = 3$: $H_3 = 3(3)^2 – 3(3) + 1 = 19$.
For $n = 4$: $H_4 = 3(4)^2 – 3(4) + 1 = 37$.
Now, the cumulative sum is $1 + 7 + 19 + 37 = 64$, which matches $4^3$.

Key Concept: Regular Polygons, Complete Graphs

b) 21
[Solution Description]
The $5^{th}$ term corresponds to $n = 5$. A regular polygon with $n+2 = 7$ sides has 7 sides and 7 vertices. The complete graph $K_{n+2} = K_7$ has $\frac{7 \times 6}{2} = 21$ edges. The total line segments include both the sides of the polygon and the edges of the complete graph, but we must avoid double-counting the sides of the polygon which are already part of $K_7$. Thus, the total line segments are the edges of $K_7$, which is 21.

Key Concept: Patterns in Shapes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of sides in a regular pentagon is 5, which is correct because a pentagon by definition has five sides. The reason explains that a regular pentagon has five equal-length sides and five equal angles, which is also true and directly explains why the assertion is correct. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Regular Polygons – Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a regular pentagon has 5 sides and 5 corners, which is correct because a pentagon is defined as a five-sided polygon. The reason states that in regular polygons, the number of sides is always equal to the number of corners, which is also correct as per the syllabus. Moreover, the reason correctly explains the assertion since it provides the general property that justifies the specific case mentioned in the assertion.

Key Concept: Regular Polygons, Perimeter and Side Length

b) 6 cm
[Solution Description]
A regular octagon has 8 equal sides. If the perimeter is 48 cm, the length of one side can be calculated by dividing the perimeter by the number of sides:
$\text{Side length} = \frac{\text{Perimeter}}{\text{Number of sides}} = \frac{48 \text{ cm}}{8} = 6 \text{ cm}$
So, the length of one side is 6 cm.

Key Concept: Complete Graphs

b) $K_5$
[Solution Description]
The sequence of complete graphs follows the order of increasing number of vertices.
Step 1: Observe the given sequence:
– $K_2$ (2 vertices)
– $K_3$ (3 vertices)
– $K_4$ (4 vertices)
The next graph should have 5 vertices, which is $K_5$.

Key Concept: Complete Graphs, Numerical Patterns

c) 6
[Solution Description]
The number of edges in a complete graph $K_n$ is given by the formula $\frac{n(n-1)}{2}$. Here, the number of edges is 15. So, we can write the equation as:
$\frac{n(n-1)}{2} = 15$
Multiply both sides by 2 to eliminate the denominator:
$n(n-1) = 30$
Expand the left side:
$n^2 – n – 30 = 0$
Factorize the quadratic equation:
$(n – 6)(n + 5) = 0$
This gives two solutions: $n = 6$ and $n = -5$. Since the number of vertices cannot be negative, the correct value is $n = 6$.

Key Concept: Complete Graphs, Numerical Patterns

d) $K_6$
[Solution Description]
First, calculate the sum of the first 5 natural numbers:
$1 + 2 + 3 + 4 + 5 = 15$
Now, find the number of vertices $n$ for which $K_n$ has 15 edges. Using the formula $\frac{n(n-1)}{2}$:
$\frac{n(n-1)}{2} = 15$
Solving as before, we get $n = 6$. Therefore, the required graph is $K_6$.

Key Concept: Stacked squares, Pattern recognition

c) 36
[Solution Description]
The sequence of stacked squares follows the pattern of perfect squares. For the $n^{th}$ shape, the number of squares is given by the formula $S_n = n^2$.
For the 6th shape:
Step 1: Substitute n = 6 into the formula.
Step 2: Calculate $S_6 = 6^2 = 36$.
Thus, the 6th shape will have 36 squares.

Key Concept: Triangular Numbers

c) 4
[Solution Description]
The first triangular number is 1 and the second is 3. Their sum is simply $1 + 3 = 4$.

Key Concept: Stacked Triangles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of little triangles in the $n^{th}$ shape of the sequence of Stacked Triangles equals the $n^{th}$ triangular number. This is true because the sequence of Stacked Triangles follows the pattern of triangular numbers, where each new row adds one more triangle than the previous row. For example, the first shape has 1 triangle, the second has 3 (1+2), the third has 6 (1+2+3), and so on.
The reason provides the formula for the $n^{th}$ triangular number, which is correct. The formula $T_n = \frac{n(n + 1)}{2}$ accurately represents the sum of the first $n$ natural numbers, matching the count of triangles in the $n^{th}$ stacked triangle shape.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Koch Snowflake iterations, Number sequence patterns

c) $\frac{64}{9}$ units
[Solution Description]
Initial perimeter P = 3 units (since equilateral triangle with side 1). Each iteration multiplies perimeter by $\frac{4}{3}$. After 1st iteration: $3 \times \frac{4}{3} = 4$. After 2nd iteration: $4 \times \frac{4}{3} = \frac{16}{3}$. After 3rd iteration: $\frac{16}{3} \times \frac{4}{3} = \frac{64}{9}$ units.

Key Concept: Koch Snowflake area, Geometric series

b) $\frac{3A}{5}$
[Solution Description]
Each iteration adds new triangles that are $\frac{1}{9}$ the area of previous triangles. The series is $\frac{A}{3} + \frac{4A}{27} + \frac{16A}{243} + …$ which is a geometric series with sum $\frac{\frac{A}{3}}{1-\frac{4}{9}} = \frac{3A}{5}$.

Key Concept: Koch Snowflake iteration, Number sequence relation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of line segments in the $n^{th}$ iteration follows $3 \times 4^{(n-1)}$. This is correct because:
– In the first iteration (n=1), it has 3 sides: $3 \times 4^{(0)} = 3$.
– In the second iteration (n=2), each side splits into 4, so $3 \times 4^{(1)} = 12$.
– Similarly, for n=3, the count becomes $3 \times 4^{(2)} = 48$, and so on.
The reason explains the mechanism behind this pattern: every prior line segment is replaced by four new ones. This directly justifies the formula in the assertion.
Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Regular Polygons and Number Sequences

c) 7
[Solution Description]
A regular heptagon is a 7-sided polygon. According to the sequence of regular polygons, the number of sides follows the counting numbers starting from 3 (triangle), 4 (quadrilateral), up to n sides. For a heptagon, the number of sides is 7.

Key Concept: Relation to Number Sequences

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of sides in a regular polygon follows the sequence of counting numbers starting from 3. This is true because, for example, a triangle has 3 sides, a quadrilateral has 4 sides, and so on. The reason states that this happens because regular polygons have equal-length sides and equal angles. While this property is true for regular polygons, it does not directly explain why the number of sides follows the counting numbers. Instead, the sequence arises by definition as we incrementally add one more side to form the next polygon.

Key Concept: Relationship Between Sides and Corners in Regular Polygons

b) Number of sides equals the number of corners
[Solution Description]
In any regular polygon, the number of sides equals the number of corners because each side connects two adjacent corners without overlapping. For example:
– Triangle has $3$ sides and $3$ corners.
– Square has $4$ sides and $4$ corners.
Therefore, the number of sides and corners are always equal.

Key Concept: Regular Polygons

c) 7
[Solution Description]
A regular heptagon is a 7-sided polygon. According to the syllabus, the number of sides in regular polygons follows the counting numbers starting from 3. The sequence for regular polygons is: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), etc. Therefore, a regular heptagon has 7 sides.

Key Concept: Koch Snowflake

c) 12
[Solution Description]
The Koch Snowflake sequence follows the pattern $3, 12, 48, …$, which can be written as $3 \times 4^{(n-1)}$ where $n$ is the position in the sequence. For the second shape ($n = 2$), the number of line segments is $3 \times 4^{(2-1)} = 3 \times 4 = 12$.

Key Concept: Regular Polygons, Koch Snowflake

c) 96
[Solution Description]
First, find the 5th term of the Koch Snowflake sequence. The sequence is $3 \times 4^{(n-1)}$. For $n = 5$, it is $3 \times 4^{4} = 3 \times 256 = 768$. Next, find the number of sides of the 6th regular polygon. The sequence starts at 3 and increases by 1 each time, so the 6th term is $3 + 5 = 8$. Now divide the 5th term of the Koch Snowflake by 8: $768 / 8 = 96$.

Key Concept: Regular polygons naming

c) Pentagon
[Solution Description]
Regular polygons are named based on the number of sides they have. A 5-sided polygon is called a pentagon. The names are derived from Greek numerical prefixes: “penta-” means five.

Key Concept: Regular Polygons and Number of Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a regular hexagon has 6 sides, which is true as per the definition of a hexagon. The reason states that the number of sides in regular polygons follows the counting numbers starting from 3 (i.e., 3, 4, 5, 6, …). This is also true since a hexagon is part of this sequence (6 sides). Moreover, the reason correctly explains why the assertion is true because the number of sides in a regular hexagon is derived from this sequence. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Regular Polygons, Number Sequences

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sequence of regular polygons starts with a triangle (3 sides) and each subsequent polygon adds one more side, giving sides as counting numbers 3, 4, 5, etc. This is true because constructing a new regular polygon involves adding a side and angle while keeping all sides and angles equal. The reason explains why this happens: each new polygon is built by adding one more side and angle to the previous one, preserving regularity. Thus, both assertation and reason are true, and the reason correctly explains the assertion.

Key Concept: Complete Graphs and Triangular Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of lines in $K_n$ corresponds to the $(n-1)$-th triangular number. This is true because triangular numbers are given by the series $1, 3, 6, 10, \dots$, which match the line counts in $K_2, K_3, K_4, \dots$. The reason provides the correct formula $\frac{n(n-1)}{2}$ for the number of lines in $K_n$, which is indeed the formula for the $(n-1)$-th triangular number. Thus, the reason correctly explains the assertion.

Key Concept: Complete Graphs and Triangular Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A complete graph $K_n$ has $n$ vertices, and each vertex is connected to every other vertex by a unique line. The total number of lines is the number of ways to choose 2 vertices out of $n$, which is given by the combination formula $C(n, 2) = \frac{n(n-1)}{2}$. This formula matches the triangular number sequence, as each triangular number represents the sum of the first $(n-1)$ natural numbers. Therefore, both Assertion and Reason correctly describe the relationship between complete graphs and triangular numbers, and Reason explains why Assertion is true.

Key Concept: Triangular Number Formula

c) $\frac{n(n-1)}{2}$
[Solution Description]
The correct formula for the number of lines in a complete graph $Kn$ is derived from triangular numbers and is given by:
$\text{Number of lines} = \frac{n(n-1)}{2}.$
This formula calculates the total connections between $n$ vertices without repetition.

Key Concept: Stacked Squares, Square Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let’s verify the assertion and reason step by step.
1) For $n = 1$: The first shape has 1 little square, which is the first odd number. Sum = 1 = $1^2$.
2) For $n = 2$: The second shape adds 3 more squares (the next odd number), making total squares = 1 + 3 = 4 = $2^2$.
3) For $n = 3$: The third shape adds 5 squares, making total squares = 1 + 3 + 5 = 9 = $3^2$.
From this pattern, it is clear that the number of little squares in the $n$-th shape is the sum of the first $n$ odd numbers, which mathematically equals $n^2$. This also matches the geometric interpretation where $n^2$ represents the area of a square with side length $n$, formed by stacking the smaller squares. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Stacked Squares, Algebraic Reasoning

b) 25
[Solution Description]
The pattern here is cumulative addition of consecutive odd numbers: Layer 1: 1 square (Total: 1). Layer 2: 3 squares (Total: 1 + 3 = 4). Layer 3: 5 squares (Total: 4 + 5 = 9). However, the question states the total after 3 layers is 14, indicating a modified pattern. We can deduce that the base layer has 2 squares (instead of 1), and each subsequent layer adds the next odd number: Layer 1: 2 squares (Total: 2). Layer 2: 2 + 3 = 5 squares (Total: 5). Layer 3: 5 + 5 = 10 squares (Total: 10). But this doesn’t match the given total of 14. Alternatively, if we consider the pattern as starting with 2 squares and then adding consecutive even numbers: Layer 1: 2 squares (Total: 2). Layer 2: 2 + 4 = 6 squares (Total: 6). Layer 3: 6 + 6 = 12 squares (Total: 12). Still not matching 14. Therefore, we must assume the pattern starts with 1 square and adds consecutive odd numbers but with an initial offset: Layer 1: 1 + 1 = 2 squares (Total: 2). Layer 2: 2 + 3 = 5 squares (Total: 5). Layer 3: 5 + 5 = 10 squares (Total: 10). This still doesn’t match. Given the inconsistency, we’ll proceed with the standard odd-number addition but adjust to meet the given total: Suppose the first three terms are 2, 5, 14. Then the next differences are +3, +9, suggesting a multiplication factor. Continuing this pattern: Layer 4: 14 + 15 = 29 squares. Layer 5: 29 + 25 = 54 squares. However, none of the options match this. Alternatively, considering the problem might have a typo, we’ll use the standard odd-number addition pattern for 5 layers: 1 + 3 + 5 + 7 + 9 = 25 squares.

Key Concept: Stacked Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The sequence of Stacked Squares shows shapes with 1, 4, 9, 16, etc., little squares. These numbers correspond to $1^2$, $2^2$, $3^2$, $4^2$, etc. This means for the $n^{th}$ shape, the number of little squares is $n^2$. The reason correctly explains that each new layer adds a square number of little squares, forming a perfect square pattern. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Visualizing Triangular Numbers

c) 10
[Solution Description]
In the stacked triangles sequence, the nth shape has a total number of little triangles equal to the nth triangular number. The 4th triangular number is calculated by adding the first 4 natural numbers:
$1 + 2 + 3 + 4 = 10$.
So, there are 10 little triangles in the 4th shape.

Key Concept: Triangular Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sum of the first $n$ natural numbers gives the $n^{th}$ triangular number. This is true because the formula for the $n^{th}$ triangular number is $T_n = \frac{n(n+1)}{2}$, which is derived from summing the first $n$ natural numbers. The reason explains that triangular numbers correspond to the count of little triangles in stacked triangles, where each row increases by one triangle. This is also true as it visually represents the concept of triangular numbers. Furthermore, the reason correctly explains why the assertion holds because the stacking pattern directly leads to the summation formula.

Key Concept: Triangular Number Sequence

c) 15
[Solution Description]
To find the next triangular number after 10, we add the next counting number to it. The pattern is as follows:
1 (first term),
1 + 2 = 3 (second term),
3 + 3 = 6 (third term),
6 + 4 = 10 (fourth term).
Therefore, the fifth term is:
10 + 5 = 15.
The next triangular number is 15.

Key Concept: Patterns in Nature

a) The arrangement of leaves on a stem
[Solution Description]
Patterns in nature include things like the arrangement of leaves, petals on flowers, or the stripes on a zebra. These are natural occurrences that follow a regular, repeated form.

Key Concept: Mathematics as an Art and Science

c) It combines creativity in pattern discovery with logical explanations
[Solution Description]
Mathematics is considered an art because it involves creativity in discovering patterns, and a science because it seeks explanations for those patterns. Both aspects are essential to the field.

Key Concept: Patterns in Numbers, Patterns in Shapes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the number of diagonals in a regular $n$-sided polygon:
1. A regular polygon has $n$ vertices.
2. From any one vertex, you can draw a line to $(n-3)$ other vertices (excluding itself and its two adjacent vertices, as those would form sides, not diagonals).
3. Since each diagonal is counted twice (once from each end), the total number of diagonals is $\frac{n(n-3)}{2}$.
Thus, both the Assertion and Reason are correct, and the Reason explains the Assertion.

Key Concept: Square and Odd Number Relationship

b) $\{1, 3, 5, 7, 9, 11\}$
[Solution Description]
The relationship between square numbers and odd numbers is that the sum of the first $n$ odd numbers gives the $n$-th square number. For example:
$1 = 1^2,$
$1 + 3 = 2^2,$
$1 + 3 + 5 = 3^2,$
$1 + 3 + 5 + 7 = 4^2,$
$1 + 3 + 5 + 7 + 9 = 5^2,$
$1 + 3 + 5 + 7 + 9 + 11 = 6^2 = 36.$
Thus, the set of odd numbers $\{1, 3, 5, 7, 9, 11\}$ sums to 36.

Key Concept: Triangular Numbers

d) 28
[Solution Description]
The sequence of triangular numbers is $1, 3, 6, 10, 15, 21, 28, \ldots$. To find the next triangular number after 21, we add the next counting number to 21. The position of 21 in the sequence corresponds to adding numbers from 1 to 6:
$1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15, 15 + 6 = 21, 21 + 7 = 28.$
Therefore, the next triangular number after 21 is 28.

Key Concept: Triangular Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the triangular number sequence is formed by adding consecutive counting numbers (1, 1+2=3, 3+3=6, etc.), which is correct. This matches the definition of triangular numbers. The reason explains that these numbers correspond to dots forming equilateral triangles, which is also true. Moreover, the reason directly explains why the assertion holds since adding consecutive numbers builds up the triangular dot pattern.

Key Concept: Triangular Numbers

c) 21
[Solution Description]
To find the next triangular number after 15, we need to understand the pattern of triangular numbers. Each triangular number is formed by adding the next natural number to the previous triangular number. For example:
1 (first term),
1 + 2 = 3 (second term),
3 + 3 = 6 (third term),
6 + 4 = 10 (fourth term),
10 + 5 = 15 (fifth term).
The next term would be 15 + 6 = 21.
Therefore, the next triangular number after 15 is 21.

Key Concept: Visualising Number Sequences

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sequence 1, 3, 6, 10, 15, … represents triangular numbers. This is correct because each term in this sequence matches the number of dots required to form equilateral triangles of increasing size. For example, 1 dot forms a single-row triangle, 3 dots form a two-row triangle, and so on. The reason explains why these numbers are called triangular numbers, which is because they correspond to the dot arrangements in equilateral triangles. Therefore, the reason correctly explains the assertion.

Key Concept: Triangular Numbers, Square Numbers

c) 1225
[Solution Description]
The next number that is both triangular and square after 36 can be found by solving for $n$ where $\frac{n(n+1)}{2} = k^2$. The known sequence of such numbers starts as 1, 36, 1225, … The next number is 1225.
Step-by-Step:
1. Triangular numbers are given by $T_n = \frac{n(n+1)}{2}$.
2. Square numbers are perfect squares $k^2$.
3. Solve $\frac{n(n+1)}{2} = k^2$ for integer solutions.
4. The next solution after $n=8$ ($T_8=36$) is $n=49$ ($T_{49}=1225$).