Grade 6 Patterns in Mathematics Quiz - Set 3 | Concept of Maths

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) $6n$

B) $6n - 2$

C) $4n + 2$

D) $5n + 1$

Answer

Correct Answer: C

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?

A) $72$ cm

B) $77$ cm

C) $82$ cm

D) $85$ cm

Answer

Correct Answer: B

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.

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) $4s + 36k$

B) $4s + 40k$

C) $10s + 10k$

D) $4s + 9k$

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) $50$

B) $500$

C) $600$

D) $625$

Answer

Correct Answer: D

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) $x + 60$

B) $x + 12$

C) $x + 55$

D) $5x + 12$

Answer

Correct Answer: C

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) $4n - 2$

B) $n^2 + n$

C) $n^2 - 1$

D) $2n + 4$

Answer

Correct Answer: B

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) $21$

B) $24$

C) $25$

D) $28$

Answer

Correct Answer: D

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) $100$

B) $104$

C) $92$

D) $108$

Answer

Correct Answer: A

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) $29$

B) $30$

C) $31$

D) $27$

Answer

Correct Answer: B

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) $49995$

B) $499995$

C) $499995$

D) $499955$

Answer

Correct Answer: C

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: A

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 $= 4 \times \text{side} = 4(s + 9k) = 4s + 36k$.

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 3) | Concept of Maths

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 3) | Concept of Maths

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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 $99999 \times 5$?

Grade 6 Patterns in Mathematics Quiz - Set 3 | Concept of Maths

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?

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 3) | Concept of Maths

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 3) | Concept of Maths

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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 $99999 \times 5$?