Class 6 Mathematics Chapter 1 Patterns in Mathematics

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: Patterns in Mathematics

c) Mathematics searches for patterns and explains why they exist
[Solution Description]
The syllabus states that mathematics is largely the search for patterns and their explanations. This means mathematicians look for regularities and understand why they occur.

Key Concept: Economic Applications

d) \$25000
[Solution Description]
Linear growth means the increase is constant over time. Here, sales increase by \$5000 per month for 5 months. To find the total increase, multiply the monthly increase by the number of months: $5000 \times 5 = \$25000$.

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 Nature and Mathematics

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the motion of stars and planets follows a predictable pattern, which is true as astronomy observations confirm regular celestial movements. The Reason explains that understanding these patterns led to the theory of gravitation, which is also true and directly explains why the Assertion is scientifically significant.

Key Concept: Patterns in Nature, Mathematics in Technology

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Fibonacci sequence is observed in various natural phenomena such as the arrangement of leaves on a stem, the branching of trees, and the spirals of shells. This pattern emerges due to its ability to optimize space and resource distribution, leading to efficient growth and minimal energy wastage. The reason correctly explains the assertion by linking the mathematical property of the Fibonacci sequence (optimal arrangement) to its biological benefits (efficiency in growth and packing).

Key Concept: Mathematical Creativity

b) It combines creativity with logical reasoning
[Solution Description]
Mathematics involves the search for patterns and their explanations, which requires logical reasoning (science). At the same time, discovering these patterns can be a creative and artistic process (art). Therefore, mathematics is seen as both.

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: Weather Prediction, Consumer Behavior

c) Modeling repeating fluctuations over time with waveform mathematics.
[Solution Description]
Both fields use periodic functions: weather models employ sine waves for cyclical climate data (e.g., annual temperature changes), while retail tracks repeating buying trends (e.g., holiday sales spikes) to optimize inventory—each leveraging time-dependent pattern analysis.

Key Concept: Mathematical patterns in real-life applications

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Fibonacci sequence is observed in various natural phenomena, including the arrangement of leaves around a stem (phyllotaxis). This pattern helps plants optimize sunlight absorption and minimize overlapping of leaves, improving photosynthetic efficiency. Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Fibonacci sequence appears in leaf arrangements.

Key Concept: Chemistry Fundamentals

d) Helium
[Solution Description]
Noble gases are elements in Group 18 of the periodic table that are odorless, colorless, and have very low chemical reactivity due to their complete valence electron shells. Helium is one of the noble gases.

Key Concept: Application of Mathematics

b) By enabling encrypted messaging
[Solution Description]
Mathematics plays a crucial role in technological advancements, especially in communication. It enables data compression algorithms for faster internet, error-correcting codes for accurate data transmission, and encryption for secure messaging. The development of mobile phones, computers, and the internet relies heavily on mathematical principles. Among the options provided, enabling encrypted messaging is a clear example of how mathematics enhances communication technology.

Key Concept: Generic Economics and Infrastructure

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that infrastructure investment contributes to economic growth, which is true as infrastructure supports trade, connectivity, and efficiency. The reason explains that better infrastructure improves productivity and lowers transaction costs, logically justifying the assertion.

Key Concept: Externalities, Infrastructure Pricing

c) \$6 million
[Solution Description]
Deadweight loss arises when social benefits (\$10M) are not fully internalized. Operator captures 40%: \$4M.
Uncaptured benefit: \$10M – \$4M = \$6M. This represents the deadweight loss.

Key Concept: Basic Engineering

c) Ohm
[Solution Description]
The ohm (Omega) is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm.

Key Concept: Basic Medicine

b) Sphygmomanometer
[Solution Description]
A sphygmomanometer is the standard device used to measure blood pressure in clinical settings.

Key Concept: Triangular and Square 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 equals the $n^{th}$ square number. This is true, as the sum of the first $n$ odd numbers can be represented by the formula:
$1 + 3 + 5 + \dots + (2n – 1) = n^2$
For example:
– When $n = 1$: $1 = 1^2$
– When $n = 2$: $1 + 3 = 4 = 2^2$
– When $n = 3$: $1 + 3 + 5 = 9 = 3^2$
The reason explains that each new odd number added forms an $L$-shape that completes the next perfect square. This is also true because adding an odd number of dots in an $L$-shape to the previous square pattern indeed constructs the next larger square.

Key Concept: Patterns in Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The given sequence is $1, 3, 5, 7, \dots$, which clearly follows the pattern of odd numbers. Odd numbers are defined as numbers not divisible by 2. The difference between consecutive terms in this sequence is 2. Therefore, after 7, the next term will be $7 + 2 = 9$. The assertion correctly identifies 9 as the next number. The reason accurately states that the sequence represents odd numbers, which is indeed a correct explanation for the assertion.

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: Sum of Odd Numbers

d) 16
[Solution Description]
Adding the first four odd numbers: $1 + 3 = 4$, $4 + 5 = 9$, $9 + 7 = 16$. The sum is 16.
Calculation: $1 + 3 + 5 + 7 = 16$.

Key Concept: Powers of 2

c) 128
[Solution Description]
The powers of 2 sequence is formed by multiplying the previous term by 2.
Given sequence: $1, 2, 4, 8, 16, 32, 64$ (up to $2^6$)
To find $2^7$: Multiply $64$ by $2$: $64 \times 2 = 128$
So, $2^7 = 128$.

Key Concept: Counting Numbers

b) 6
[Solution Description]
The given sequence is the counting numbers, where each subsequent number increases by 1. The last number provided is 5, so the next number in the sequence will be $5 + 1 = 6$.

Key Concept: Counting numbers, Visual patterns

b) 55
[Solution Description]
The problem involves counting the number of dots in a triangular pattern. Each row $n$ has $n$ dots. To find the total number of dots from row 1 to row 10, we need to calculate the sum of the first 10 counting numbers. The formula for the sum of the first $n$ counting numbers is $\frac{n(n + 1)}{2}$. Here, $n = 10$.
Step 1: Plug in $n = 10$ into the formula: $\frac{10(10 + 1)}{2} = \frac{10 \times 11}{2} = \frac{110}{2} = 55$.
So, the total number of dots is 55.

Key Concept: Number patterns

c) 16
[Solution Description]
This pattern represents square numbers:
1 = $1^2$ = 1
1+2+1 = 4 = $2^2$
1+2+3+2+1 = 9 = $3^2$
The next term would be:
1+2+3+4+3+2+1 = 16 = $4^2$
Therefore, the next number is $16$.

Key Concept: Sum of Odd Numbers

c) 25
[Solution Description]
The first 5 odd numbers are 1, 3, 5, 7, and 9. The sum can be calculated as follows:
Step 1: Add the first two odd numbers: $1 + 3 = 4$
Step 2: Add the next odd number to the previous sum: $4 + 5 = 9$
Step 3: Add the next odd number to the previous sum: $9 + 7 = 16$
Step 4: Add the last odd number to the previous sum: $16 + 9 = 25$
Therefore, the sum of the first 5 odd numbers is $25$.

Key Concept: Sum of Odd Numbers Pattern

c) 9
[Solution Description]
The first 3 odd numbers are $1$, $3$, and $5$. Their sum is calculated as follows: First, add $1 + 3 = 4$. Then, add the result to the next odd number: $4 + 5 = 9$. According to the pattern, the sum of the first $n$ odd numbers equals $n^2$, so for $n=3$, $3^2 = 9$.

Key Concept: Even numbers, Patterns

c) 310
[Solution Description]
The first $n$ even numbers form a sequence: $2, 4, 6, …, 2n$. The sum of the first $n$ even numbers is given by $S = n(n + 1)$.
Step 1: Calculate the sum of the first $10$ even numbers.
$S_{10} = 10 \times (10 + 1) = 110$
Step 2: Calculate the sum of the first $20$ even numbers.
$S_{20} = 20 \times (20 + 1) = 420$
Step 3: Subtract $S_{10}$ from $S_{20}$ to get the required result.
$420 – 110 = 310$

Key Concept: Even numbers, Word problem

c) 156
[Solution Description]
The number of apples in each row follows the sequence of even numbers: $2, 4, 6, …, 2n$, where $n$ is the row number.
Step 1: The total number of apples is the sum of the first $12$ even numbers.
$S_{12} = 12 \times (12 + 1) = 156$

Key Concept: Sum of Triangular Numbers

b) 100
[Solution Description]
This problem involves adding numbers from 1 up to 10 and then back down to 1. The sum can be broken into two parts: the sum of numbers from 1 to 10 and the sum of numbers from 9 back to 1.
First, calculate the sum of numbers from 1 to 10 using the formula for triangular numbers:
$S_{10} = \frac{10(10 + 1)}{2} = 55$
Next, calculate the sum of numbers from 9 back to 1:
$S_9 = \frac{9(9 + 1)}{2} = 45$
Now add both sums:
$55 + 45 = 100$
Therefore, the total sum is 100.

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]
First, find the 4th and 5th triangular numbers. The $n^{th}$ triangular number is given by $T_n = \frac{n(n+1)}{2}$.
For $n = 4$:
$T_4 = \frac{4(4+1)}{2} = \frac{20}{2} = 10$.
For $n = 5$:
$T_5 = \frac{5(5+1)}{2} = \frac{30}{2} = 15$.
Now, add them: $T_4 + T_5 = 10 + 15 = 25$. Since $25$ is $5^2$, it is a perfect square.
Next, check if this holds for any two consecutive triangular numbers $T_n$ and $T_{n+1}$:
$T_n + T_{n+1} = \frac{n(n+1)}{2} + \frac{(n+1)(n+2)}{2} = \frac{n(n+1) + (n+1)(n+2)}{2}$.
Factor out $(n+1)$:
$(n+1)\left(\frac{n + (n+2)}{2}\right) = (n+1)\left(\frac{2n+2}{2}\right) = (n+1)(n+1) = (n+1)^2$, which is always a perfect square.
Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Square Numbers

c) 16
[Solution Description]
The square of a number means multiplying the number by itself. So to find the square of 4, we calculate:
$4 \times 4 = 16$
Therefore, the square of 4 is 16.

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 application

b) 7
[Solution Description]
In a Virahānka sequence, each subsequent number is the sum of the two preceding numbers. Given the first two numbers as $3$ and $4$, the third number is $3 + 4 = 7$.

Key Concept: Virahānka numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Virahānka sequence follows the pattern where each term is the sum of the two preceding terms. So, after 8 and 13, the next term is $8 + 13 = 21$. Therefore, both Assertion (A) and Reason (R) are true, and Reason correctly explains Assertion because the sequence rule justifies why 21 is the next term.

Key Concept: Counting Numbers

b) 1, 2, 3, 4, 5
[Solution Description]
Counting numbers are the simplest sequence of numbers starting from 1 and increasing by 1 each time. The sequence from 1 to 5 is $1, 2, 3, 4, 5$.

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 sequence $1, 3, 6, 10, 15$ represents triangular numbers as they can be arranged into equilateral triangles, which is true. The reason explains that triangular numbers are formed by adding consecutive natural numbers (e.g., $1 = 1$, $3 = 1+2$, $6 = 1+2+3$, etc.), which is also true and correctly explains why the given numbers are called triangular numbers.

Key Concept: Square Numbers

c) 36
[Solution Description]
Square numbers are formed by multiplying a number by itself. The sequence follows $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$. The next number in the sequence is $6^2 = 36$. Therefore, the number of dots needed is $36$.

Key Concept: Hexagonal Numbers, Visual Representation of Number Sequences

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let’s verify the assertion: The given sequence (1, 7, 19, 37, …) indeed matches the initial terms of hexagonal numbers when visualized. For example:
1 dot forms a trivial hexagon.

7 dots form a hexagon with one central dot and six surrounding dots.

19 dots form a larger hexagon with an additional outer layer.

Now, check the reason: The general formula for hexagonal numbers is $H_n = 3n^2 – 3n + 1$. Let’s test it for n=1, 2, 3, and 4:

For n=1: $H_1 = 3(1)^2 – 3(1) + 1 = 1$ (matches the first term).

For n=2: $H_2 = 3(4) – 6 + 1 = 7$ (matches the second term).

For n=3: $H_3 = 3(9) – 9 + 1 = 19$ (matches the third term).

For n=4: $H_4 = 3(16) – 12 + 1 = 37$ (matches the fourth term).

Thus, both the assertion and reason are true. Furthermore, the formula in the reason justifies why these numbers represent hexagonal arrangements, making it the correct explanation for the assertion.

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: Patterns in Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sequence $1, 3, 6, 10, 15, 21, \ldots$ increases by consecutive integers starting from 2. Let’s verify this:
– The difference between 3 and 1 is 2.
– The difference between 6 and 3 is 3.
– The difference between 10 and 6 is 4.
– This pattern continues, confirming the assertion is true.
The reason states that triangular numbers are formed by adding natural numbers sequentially. Indeed, triangular numbers are sums of the first $n$ natural numbers:
– $1 = 1$
– $1 + 2 = 3$
– $1 + 2 + 3 = 6$
– $1 + 2 + 3 + 4 = 10$, etc.
The reason correctly explains why the sequence in the assertion follows the described pattern. Hence, both the assertion and reason are true, and the reason is the correct explanation.

Key Concept: Sum of consecutive odd numbers, Dot grid patterns

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason:
Step 1: Calculate the sum of the first 15 odd numbers.
The first 15 odd numbers are: $1, 3, 5, \dots, 29$.
Their sum is given by:
$S = 1 + 3 + 5 + \dots + 29$
Step 2: Use the formula for the sum of the first $n$ odd numbers, which is $S = n^2$.
For $n = 15$, this gives:
$S = 15^2 = 225$
Step 3: Count the number of dots in a $15 \times 15$ square grid.
For a square grid with side length $n$, the total number of dots is $n^2$. So:
$15 \times 15 = 225$
Both the sum of the first 15 odd numbers and the number of dots in a $15 \times 15$ grid are equal to 225. Thus, both Assertion (A) and Reason (R) are true. Moreover, Reason (R) explains why Assertion (A) is true because it provides the general formula that connects the sum of odd numbers to square grid dots.

Key Concept: Square Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 16 can be represented as a 4$\times$4 dot grid, which is true because $4 \times 4 = 16$. The reason states that square numbers are the sum of consecutive odd numbers starting from 1. This is also true because $1 + 3 + 5 + 7 = 16$, and this pattern holds for all square numbers. Moreover, the reason correctly explains why 16 is a square number, as it follows the property of being the sum of consecutive odd numbers.

Key Concept: Triangular Numbers, Properties of 6T_n + 1

b) Centered hexagonal numbers
[Solution Description]
For $n=1$, $T_1 = 1$, so $6 \times 1 + 1 = 7$. For $n=2$, $T_2 = 3$, so $6 \times 3 + 1 = 19$. For $n=3$, $T_3 = 6$, so $6 \times 6 + 1 = 37$. These numbers follow the pattern of centered hexagonal numbers: $3n(n-1) + 1$.

Key Concept: Adding Consecutive Triangular Numbers

b) Square numbers
[Solution Description]
Adding consecutive triangular numbers gives square numbers.
Let’s verify with examples:
1 (first triangular) + 3 (second triangular) = 4 (square of 2)
3 (second triangular) + 6 (third triangular) = 9 (square of 3)
6 (third triangular) + 10 (fourth triangular) = 16 (square of 4)
Hence, the sum of consecutive triangular numbers results in square numbers.

Key Concept: Cube Numbers

c) 64
[Solution Description]
To find the total number of small cubes in a larger cube with side length 4, we use the formula for cube numbers: $n^3$, where $n$ is the side length. Here, $n = 4$.
Step 1: Calculate $4^3$.
$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$
So, there are 64 small cubes in total.

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: Relations Among Number Sequences, Sum of Odd Numbers and Square Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The sum of the first $n$ odd numbers can be represented as:
$1 + 3 + 5 + \dots + (2n – 1)$
To find the sum, note that this is an arithmetic series with the first term $a_1 = 1$, last term $a_n = 2n – 1$, and number of terms $n$. The sum $S_n$ of an arithmetic series is given by:
$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(1 + (2n – 1)) = \frac{n}{2}(2n) = n^2$
This proves the Assertion (A). The Reason (R) provides a visual explanation: each odd number represents an L-shaped border around the previous square, forming a new larger square. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Sum of Odd Numbers Pattern

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 square number. This is verified by the pattern shown in the syllabus:
$1 = 1$ (which is $1^2$),
$1 + 3 = 4$ (which is $2^2$),
$1 + 3 + 5 = 9$ (which is $3^2$), and so on.
The reason explains that this happens because each addition of an odd number forms a new layer in a square grid of dots, resulting in a larger square. For example:
– A $1 \times 1$ square has $1$ dot.
– Adding $3$ dots around it forms a $2 \times 2$ square with $4$ dots.
– Adding $5$ dots around that forms a $3 \times 3$ square with $9$ dots.
Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Sum of odd numbers

c) 8
[Solution Description]
We know sum of first n odd numbers = $n^2$.
Given $n^2$ = 64.
Therefore n = $\sqrt{64}$ = 8.

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: Large Symmetrical Addition

c) 2601
[Solution Description]
The highest number in this sequence is 51. According to the pattern, the sum of such a symmetrical sequence is equal to the square of the highest number. Therefore:
Calculation:
$1 + 2 + \ldots + 50 + 51 + 50 + \ldots + 2 + 1 = 51^2 = 2601$

Key Concept: Adding Triangular Numbers

b) 16
[Solution Description]
To find the square number obtained by adding the triangular numbers $6$ and $10$, follow these steps:
1. The problem states that adding two consecutive triangular numbers gives a square number.
2. Here, $6$ and $10$ are consecutive triangular numbers (3rd and 4th triangular numbers respectively).
3. Add them: $6 + 10 = 16$.
4. Since $16$ is a perfect square ($4 \times 4$), the answer is $16$.

Key Concept: Verification of Pattern

b) 36
[Solution Description]
To verify the pattern by adding the next pair of consecutive triangular numbers after $10$ and $15$:
1. The problem involves finding the next pair of consecutive triangular numbers after $10$ (4th) and $15$ (5th).
2. The next pair would be $15$ (5th) and $21$ (6th).
3. Add them: $15 + 21 = 36$.
4. Check if the result is a square number. Indeed, $36$ is $6 \times 6$.

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: Adding powers of 2, Sum pattern

a) $2^n$
[Solution Description]
The sum of the first $n$ powers of 2 is given by $S = 1 + 2 + 4 + \dots + 2^{n-1}$. This is a geometric series with sum $S = 2^n – 1$. Adding 1 to this sum gives $S + 1 = 2^n – 1 + 1 = 2^n$. Therefore, the final number is the next power of 2 after the last term in the series.

Key Concept: Multiplying Triangular Numbers by 6 and Adding 1

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The first triangular number is $T_1 = 1$. Multiplying it by 6 gives $6 \times 1 = 6$, and adding 1 results in $6 + 1 = 7$. The formula for the $n$-th triangular number is $T_n = \frac{n(n + 1)}{2}$. For $n = 1$, this correctly gives $T_1 = 1$. The reason explains how the assertion is derived using the general formula for triangular numbers.

Key Concept: Pattern recognition

d) Both can be expressed as $3n(n-1) + 1$ where n is an integer
[Solution Description]
The sequence $7, 19, 37, 61, \dots$ is related to centered hexagonal numbers. Each term fits the formula for centered hexagonal numbers, which are of the form $3n(n-1) + 1$ where $n$ starts from 2. This shows a direct connection between the two sequences.

Key Concept: Sum of Hexagonal Numbers and Cubes

b) $8$
[Solution Description]
The first hexagonal number is $1$ and the second is $7$. Their sum is $1 + 7 = 8$. According to the syllabus, this sum represents $2^3$.

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: Koch Snowflake, Powers of 4

d) 192
[Solution Description]
The number of line segments in the Koch snowflake follows the pattern $3, 12, 48, \dots$, i.e., each iteration multiplies the previous count by 4. For 3 iterations starting from 3 line segments:
1st iteration: $3 \times 4 = 12$
2nd iteration: $12 \times 4 = 48$
3rd iteration: $48 \times 4 = 192$

Key Concept: Stacked Triangles, Number Sequences

b) 55
[Solution Description]
The sequence follows the triangular numbers: $1, 3, 6, 10, \dots$. The formula for the $n^{th}$ triangular number is $T_n = \frac{n(n+1)}{2}$. For $n = 10$, $T_{10} = \frac{10 \times 11}{2} = 55$.

Key Concept: Regular Polygons, Angle Calculation

b) $128.57^\circ$
[Solution Description]
The formula to calculate the measure of each interior angle of a regular $n$-sided polygon is:
$\text{Interior angle} = \frac{(n – 2) \times 180^\circ}{n}$
For $n = 7$:
$\frac{(7 – 2) \times 180^\circ}{7} = \frac{5 \times 180^\circ}{7} = \frac{900^\circ}{7} \approx 128.57^\circ$
So, the measure of each interior angle is approximately $128.57^\circ$.

Key Concept: Sequence of Regular Polygons

d) 3, 4, 5, 6, $\dots$
[Solution Description]
Regular polygons start with a triangle (3 sides), followed by quadrilateral (4 sides), pentagon (5 sides), and so on. Therefore, the sequence of sides is the counting numbers starting from 3: $3, 4, 5, 6, \dots$

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: Complete Graphs, Number of Edges

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of edges in $K_n$ is $\frac{n(n-1)}{2}$, which is correct because in a complete graph, each vertex is connected to every other vertex exactly once. Therefore, for $n$ vertices, the number of edges is the combination of $n$ taken 2 at a time, i.e., $\frac{n(n-1)}{2}$. The reason correctly explains the assertion by stating that every pair of distinct vertices is connected by exactly one edge, justifying the formula used in the assertion.

Key Concept: Complete Graphs

d) 6
[Solution Description]
A complete graph $K_n$ has $\frac{n(n-1)}{2}$ edges. For $K_4$, $n = 4$.
Step 1: Plug $n = 4$ into the formula:
$$\text{Edges} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = 6$$
So, $K_4$ has 6 edges.

Key Concept: Stacked Squares

b) 25
[Solution Description]
To find the number of little squares in the fifth shape:
Step 1: Observe the pattern in the given sequence of little squares: 1, 4, 9, 16.
Step 2: Recognize that these numbers are perfect squares. Specifically:
$1 = 1^2$, $4 = 2^2$, $9 = 3^2$, and $16 = 4^2$.
Step 3: For the fifth shape, the number of squares will follow the same pattern, which is $5^2 = 25$.

Key Concept: Stacked triangles and squares, Pattern analysis

c) 12
[Solution Description]
The sequence alternates between triangles and squares. The pattern for triangles is triangular numbers ($T_n = \frac{n(n+1)}{2}$), and for squares, it’s perfect squares ($S_n = n^2$).
For the 4th term:
Step 1: Identify the type of shape. The sequence starts with triangle (1st term), square (2nd term), triangle (3rd term), so the 4th term will be a square.
Step 2: Determine the position in its own sequence. This is the 2nd square in the alternating sequence (since the 2nd term was the 1st square).
Step 3: Apply the square formula for n=2: $S_2 = 2^2 = 4$.
However, we need to consider the cumulative count. The question asks for the total number of shapes (triangles + squares) up to the 4th term.
Term 1: Triangle, $T_1 = 1$
Term 2: Square, $S_1 = 1^2 = 1$ (But according to problem statement, 2nd term has 4 squares)
Correction: The problem states specific counts for initial terms, not following standard formulas.
Given:
1st term: 1 triangle
2nd term: 4 squares
3rd term: 3 triangles
4th term: ?
Since the 2nd square in the sequence had 4 squares (which is $2^2$), likely the pattern continues.
Thus, 4th term would be the 2nd square in the sequence: $2^2 = 4$ squares.
Total shapes would be sum of all shapes up to 4th term: 1 (triangle) + 4 (squares) + 3 (triangles) + 4 (squares) = 12.

Key Concept: Stacked Triangles

b) 15
[Solution Description]
To find the number of little triangles in the fifth shape:
Step 1: Observe the pattern in the given sequence of little triangles: 1, 3, 6, 10.
Step 2: Recognize that the differences between consecutive terms are increasing by 1 each time:
$3 – 1 = 2$, $6 – 3 = 3$, $10 – 6 = 4$.
Step 3: The next difference should be $5$, so the next term is $10 + 5 = 15$.
Alternatively, recognize that these numbers are triangular numbers, where the $n^{th}$ triangular number is given by $\frac{n(n+1)}{2}$. For the fifth shape ($n=5$): $\frac{5(5+1)}{2} = 15$.

Key Concept: Fractal dimension, Scaling properties

b) $\frac{\log(4)}{\log(3)}$ ≈ 1.26186
[Solution Description]
Fractal dimension D is calculated using $D = \frac{\log(N)}{\log(1/r)}$. Here, $N=4$ and $r=\frac{1}{3}$, so $D = \frac{\log(4)}{\log(3)} \approx 1.26186$.

Key Concept: Koch Snowflake Multiplier

c) 4
[Solution Description]
From the syllabus, the sequence for line segments is 3, 12, 48,… To find the common ratio, divide any term by its preceding term. For example, $\frac{12}{3} = 4$ or $\frac{48}{12} = 4$. Thus, the common ratio is 4.

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: Relation to Number Sequences, Regular Polygons

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the number of sides in a regular polygon sequence corresponds to the triangular number sequence. However, this is incorrect. The number of sides in a regular polygon sequence actually corresponds to the counting numbers starting from 3 (i.e., 3, 4, 5, 6, …), not the triangular numbers (1, 3, 6, 10, …).
The reason given claims that triangular numbers represent the total number of sides needed to form nested equilateral triangles. This statement is true because for nested equilateral triangles, each additional triangle adds its sides to the total count, which follows the pattern of triangular numbers.
Since the Assertion is false and the Reason is true, the correct answer is option d).

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: Counting Lines in Complete Graphs

c) 10
[Solution Description]
The number of lines in a complete graph follows the triangular number sequence. The formula for the nth triangular number is $\frac{n(n-1)}{2}$.
– 1st graph: $\frac{1(0)}{2} = 0$ lines
– 2nd graph: $\frac{2(1)}{2} = 1$ line
– 3rd graph: $\frac{3(2)}{2} = 3$ lines
– 4th graph: $\frac{4(3)}{2} = 6$ lines
– 5th graph: $\frac{5(4)}{2} = 10$ lines
Hence, the 5th complete graph will have $10$ lines.

Key Concept: Shape sequences and number sequences

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 sequence of regular polygons follows the counting numbers starting from 3 (3, 4, 5, …). This is correct as a regular triangle has 3 sides, a square has 4 sides, and so on. The reason explains that regular polygons have equal-length sides and equal angles, which allows us to count their sides sequentially. Both the assertion and the reason are true, and the reason correctly explains why the assertion is valid because the properties of regular polygons lead to sequential side counts.

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

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: 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: 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, Number sequences

c) 15 sides
[Solution Description]
The formula for the internal angle of a regular polygon with $n$ sides is:
$\text{Internal Angle} = \frac{(n-2) \times 180^\circ}{n}$
Given the internal angle is 156 degrees:
$156 = \frac{(n-2) \times 180}{n}$
Multiply both sides by $n$ to eliminate the denominator:
$156n = 180n – 360$
Rearrange terms to solve for $n$:
$24n = 360 \implies n = 15$
The polygon has 15 sides.

Key Concept: Complete Graphs, Triangular Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the number of lines in $K_7$ using the given formula for complete graphs.
Step 1: Identify the value of $n$ for $K_7$. Here, $n = 7$.
Step 2: Apply the triangular number formula:
$\frac{n(n-1)}{2} = \frac{7(7-1)}{2}$
Step 3: Simplify inside the parentheses:
$\frac{7 \times 6}{2}$
Step 4: Perform multiplication and division:
$\frac{42}{2} = 21$
Therefore, the number of lines in $K_7$ is 21, which matches the assertion. The reason correctly provides the general formula for the number of lines in any complete graph, and it accurately explains why the assertion is true.

Key Concept: Complete Graphs and Triangular Numbers

c) 10
[Solution Description]
The number of lines in a complete graph $K_n$ is given by the formula:
$\frac{n(n-1)}{2} = 45$
Multiply both sides by 2:
$n(n-1) = 90$
Expand the equation:
$n^2 – n – 90 = 0$
Solve the quadratic equation:
$n = \frac{1 \pm \sqrt{1 + 360}}{2} = \frac{1 \pm \sqrt{361}}{2} = \frac{1 \pm 19}{2}$
Take the positive solution:
$n = \frac{20}{2} = 10$
So, the value of $n$ is 10.

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: Shape Sequence and Pattern Recognition

c) Perfect squares ($1^2$, $2^2$, $3^2$, …)
[Solution Description]
The number of little squares increases as perfect squares ($n^2$) where $n$ is the position of the term in the sequence.

Key Concept: Stacked Squares, Square Numbers

c) 25
[Solution Description]
The pattern of stacked squares follows the formula for centered square numbers: $n^2 + (n-1)^2$. For the first term ($n=1$): $1^2 + 0^2 = 1$ square. Second term ($n=2$): $2^2 + 1^2 = 4 + 1 = 5$ squares. Third term ($n=3$): $3^2 + 2^2 = 9 + 4 = 13$ squares. For the fourth term ($n=4$): $4^2 + 3^2 = 16 + 9 = 25$ squares.

Key Concept: Square Numbers

c) Square numbers
[Solution Description]
The stacked squares sequence visually represents the square numbers. Each term corresponds to a perfect square: $1, 4, 9, 16, 25, …$. Thus, the sequence is the square numbers.

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 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: Triangular numbers, Sum of consecutive triangular numbers

c) 6
[Solution Description]
The 5th triangular number is calculated as $T_5 = \frac{5 \times (5 + 1)}{2} = 15$.
The 6th triangular number is calculated as $T_6 = \frac{6 \times (6 + 1)}{2} = 21$.
The sum of the 5th and 6th triangular numbers is $15 + 21 = 36$.
Since $36$ is a perfect square ($6 \times 6 = 36$), the required integer is $6$.

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: Visualizing Number Sequences

c) 9
[Solution Description]
A regular nonagon is part of the sequence of regular polygons. The number of sides follows the counting numbers starting from 3 (triangle). The sequence is:
3 (triangle), 4 (quadrilateral), 5 (pentagon), …, 9 (nonagon).
Thus, a nonagon has 9 sides.

Key Concept: Practical Applications of Mathematical Patterns

b) Satellite technology
[Solution Description]
Understanding the patterns in the motion of stars and planets led to the theory of gravitation, which enabled advancements like satellite launches and space exploration.

Key Concept: Even Number Sequence

b) 10
[Solution Description]
The given sequence is the even number sequence. Each subsequent number increases by 2. The current last number is 8, so adding 2 gives 10.

Key Concept: Virahānka numbers, Powers of 2

c) The differences form the negated Virahānka sequence shifted by two.
[Solution Description]
The Virahānka sequence is $1, 2, 3, 5, 8, 13,\dots$. The powers of 2 sequence is $1, 2, 4, 8, 16, 32,\dots$. Subtract corresponding terms:
1st term: $1 – 1 = 0$
2nd term: $2 – 2 = 0$
3rd term: $3 – 4 = -1$
4th term: $5 – 8 = -3$
5th term: $8 – 16 = -8$
The resulting sequence is $0, 0, -1, -3, -8,\dots$, which resembles the negated Virahānka sequence shifted by two positions.

Key Concept: Square Numbers

b) 16
[Solution Description]
Square numbers are numbers like $1, 4, 9, 16, \dots$ where each is obtained by multiplying a whole number by itself (e.g., $4=2\times2$, $9=3\times3$). From the options, only 16 fits this pattern ($4\times4=16$).

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: Cube Numbers, Pictorial Representation

c) 37
[Solution Description]
The $n$-th cube number has $n^3$ dots. The difference between the 4th and 3rd cubes is $4^3 – 3^3 = 64 – 27 = 37$.
Step-by-Step:
1. Calculate the 4th cube number: $4 \times 4 \times 4 = 64$.
2. Calculate the 3rd cube number: $3 \times 3 \times 3 = 27$.
3. Subtract to find additional dots needed: $64 – 27 = 37$.

Key Concept: Visualising Number Sequences, Relations Among Number Sequences

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the sum of consecutive odd numbers starting from 1 results in a perfect square. This is true, as demonstrated by examples: $1 = 1^2$, $1 + 3 = 4 = 2^2$, $1 + 3 + 5 = 9 = 3^2$, and so on.
The Reason explains that square numbers can be visually represented by partitioning them into successive odd-numbered dot arrangements. For instance, a $2 \times 2$ square (4 dots) can be divided into 1 dot and 3 dots arranged in an L-shape, showing how adding odd numbers forms squares visually.
Both are true, and the Reason correctly explains why the sum of consecutive odd numbers yields perfect squares through their visual representation.