Question 1

1. An architect is designing a walkway using hexagonal tiles. If one hexagon requires 6 border stones, two joined hexagons require 10 border stones, and three joined hexagons require 14 border stones, which expression represents the number of stones needed for n hexagons in a row?

A) 6nB) 6n - 2C) 4n + 2D) 5n + 1

Answer

✓ Correct Answer: Option C

Explanation: Solution: Let's analyze the pattern: For n=1: Stones = 6 For n=2: Stones = 10 = 6 + 4 For n=3: Stones = 14 = 10 + 4 This is an arithmetic pattern where each step adds 4 stones. The rule is: $4 \times n + 2$. Verification: For n=1, 4(1)+2 = 6. For n=2, 4(2)+2 = 10.

Question 2

2. In a robotic laboratory, a specialized arm extends its length according to the sequence: 7 cm, 12 cm, 17 cm, 22 cm... What will be the length of the arm at the $15^{th}$ stage?

72 cm

Question 3

3. A digital artist creates a pattern where each square has a side length s. The perimeter of the first square is 4s. If the side length of each subsequent square increases by a constant k, such that the sides are s, s+k, s+2k..., what is the perimeter of the $10^{th}$ square?

A) 4s + 36kB) 4s + 40kC) 10s + 10kD) 4s + 9k

Answer

✓ Correct Answer: Option A

Explanation: Solution: Side of $1^{st}$ square = s Side of $2^{nd}$ square = s + k Side of $n^{th}$ square = s + (n-1)k Side of $10^{th}$ square = s + 9k Perimeter $= 4 \times side = 4(s + 9k) = 4s + 36k$.

Question 4

4. Observe the pattern: $1 = 1^2$ $1 + 3 = 4 = 2^2$ $1 + 3 + 5 = 9 = 3^2$ Following this logic, what is the sum of the first 25 consecutive odd numbers?

A) 50B) 500C) 600D) 625

Answer

✓ Correct Answer: Option D

Explanation: Solution: The pattern shows that the sum of the first n odd numbers is always $n^2$. Here, n = 25. Sum $= 25^2 = 25 \times 25 = 625$.

Question 5

5. A local library arranges books on shelves. The first shelf has x books. Every shelf below it has 5 more books than the shelf above it. If there are 12 shelves, how many books are on the bottom-most shelf?

A) x + 60B) x + 12C) x + 55D) 5x + 12

Answer

✓ Correct Answer: Option C

Explanation: Solution: Shelf 1: x Shelf 2: x + 5 Shelf 3: x + 10 = x + 5(2) Following the pattern, Shelf n has x + 5(n-1) books. For n=12: Books = x + 5(12-1) = x + 5(11) = x + 55.

Question 6

6. Consider the sequence of numbers: 2, 6, 12, 20, 30... Which of the following algebraic rules defines the $n^{th}$ term of this sequence?

A) 4n - 2B) $n^2 + n$C) $n^2 - 1$D) 2n + 4

Answer

✓ Correct Answer: Option B

Explanation: Solution: Let's test $n^2 + n$: If $n=1: 1^2 + 1 = 2$ If $n=2: 2^2 + 2 = 6$ If $n=3: 3^2 + 3 = 12$ If $n=4: 4^2 + 4 = 20$ The rule $n^2 + n$ (or n(n+1)) matches the sequence perfectly.

Question 7

7. A pattern of dots forms a triangle. The $1^{st}$ triangle has 1 dot, the $2^{nd}$ has 3 dots, the $3^{rd}$ has 6 dots, and the $4^{th}$ has 10 dots. How many dots will the $7^{th}$ triangle in this sequence have?

A) 21B) 24C) 25D) 28

Answer

✓ Correct Answer: Option D

Explanation: Solution: These are triangular numbers. The $n^{th}$ term is the sum of the first n natural numbers: $\frac{n(n+1)}{2}$. For n=7: Dots $= \frac{7(7+1)}{2} = \frac{7 \times 8}{2} = \frac{56}{2} = 28$.

Question 8

8. If the pattern of a mathematical operation is $a \star b = (a \times b) + (a - b)$, calculate the value of 12 \star 8.

A) 100B) 104C) 92D) 108

Answer

✓ Correct Answer: Option A

Explanation: Solution: Given rule: $a \star b = (a \times b) + (a - b)$ Substitute a = 12 and b = 8: $12 \star 8 = (12 \times 8) + (12 - 8)$ 12 \star 8 = 96 + 4 = 100.

Question 9

9. A sequence is defined as $T_n = 3n - 1$. If the value of a certain term in this sequence is 89, what is the position (n) of that term?

A) 29B) 30C) 31D) 27

Answer

✓ Correct Answer: Option B

Explanation: Solution: We are given $T_n = 89$ and $T_n = 3n - 1$. So, 3n - 1 = 89 3n = 89 + 1 3n = 90 $n = \frac{90}{3} = 30$.

Question 10

10. Look at the distributive pattern: $9 \times 5 = 45$, $99 \times 5 = 495$, $999 \times 5 = 4995$. Based on this pattern, what is the value of $99999 \times 5$?

A) 49995B) 499995C) 499995D) 499955

Answer

✓ Correct Answer: Option C

Explanation: Solution: Pattern Analysis: 1 nine: 4 (zero nines) 5 2 nines: 4 (one nine) 5 3 nines: 4 (two nines) 5 For 5 nines (99999): The result should be 4 followed by (5-1) = 4 nines, then 5. Result = 499995.

Accepted Answer

✓ Correct Answer: Option B

Explanation: Solution: The sequence is 7, 12, 17, 22... The first term a = 7. The common difference d = 12 - 7 = 5. The general rule for the $n^{th}$ term is 5n + 2. For n=15: Length = 5(15) + 2 = 75 + 2 = 77 cm.

Grade 6 Patterns in Mathematics Quiz - Set 3

1. An architect is designing a walkway using hexagonal tiles. If one hexagon requires 6 border stones, two joined hexagons require 10 border stones, and three joined hexagons require 14 border stones, which expression represents the number of stones needed for n hexagons in a row?

2. In a robotic laboratory, a specialized arm extends its length according to the sequence: 7 cm, 12 cm, 17 cm, 22 cm... What will be the length of the arm at the $15^{th}$ stage?

3. A digital artist creates a pattern where each square has a side length s. The perimeter of the first square is 4s. If the side length of each subsequent square increases by a constant k, such that the sides are s, s+k, s+2k..., what is the perimeter of the $10^{th}$ square?

4. Observe the pattern: <br> $1 = 1^2$ <br> $1 + 3 = 4 = 2^2$ <br> $1 + 3 + 5 = 9 = 3^2$ <br> Following this logic, what is the sum of the first 25 consecutive odd numbers?

5. A local library arranges books on shelves. The first shelf has x books. Every shelf below it has 5 more books than the shelf above it. If there are 12 shelves, how many books are on the bottom-most shelf?

6. Consider the sequence of numbers: 2, 6, 12, 20, 30... Which of the following algebraic rules defines the $n^{th}$ term of this sequence?

7. A pattern of dots forms a triangle. The $1^{st}$ triangle has 1 dot, the $2^{nd}$ has 3 dots, the $3^{rd}$ has 6 dots, and the $4^{th}$ has 10 dots. How many dots will the $7^{th}$ triangle in this sequence have?

8. If the pattern of a mathematical operation is $a \star b = (a \times b) + (a - b)$, calculate the value of 12 \star 8.

9. A sequence is defined as $T_n = 3n - 1$. If the value of a certain term in this sequence is 89, what is the position (n) of that term?

10. Look at the distributive pattern: $9 \times 5 = 45$, $99 \times 5 = 495$, $999 \times 5 = 4995$. Based on this pattern, what is the value of $99999 \times 5$?