Grade 6 Patterns in Mathematics Quiz (Hard) - Set 5 | Concept of Maths

Question 1

1. A galactic explorer observes a pattern of stars forming a sequence. The number of stars in the $n^{th}$ formation follows the rule $S = n^2 + n + 1$. How many stars will be in the $12^{th}$ formation?

A) 144 stars

B) 156 stars

C) 157 stars

D) 169 stars

Answer

Correct Answer: C

Solution: To find the number of stars for $n = 12$, substitute the value directly into the algebraic rule: $$S = 12^2 + 12 + 1$$ $$S = 144 + 12 + 1$$ $$S = 156 + 1 = 157$$ The $12^{th}$ stellar formation contains exactly **157 stars**.

Question 2

2. Consider a pattern where each term is the sum of the preceding two terms (starting with 1 and 2). If the sequence is $1, 2, 3, 5, 8, 13, \dots$, what is the $10^{th}$ term?

A) 55

B) 89

C) 144

D) 34

Answer

Correct Answer: B

Solution: This tracks a Fibonacci-style sequence starting deliberately with 1 and 2. Let's list the terms out to position 10: * 1st term: 1 * 2nd term: 2 * 3rd term: $1 + 2 = 3$ * 4th term: $2 + 3 = 5$ * 5th term: $3 + 5 = 8$ * 6th term: $5 + 8 = 13$ * 7th term: $8 + 13 = 21$ * 8th term: $13 + 21 = 34$ * 9th term: $21 + 34 = 55$ * 10th term: $34 + 55 = 89$.

Question 3

3. An architect designs a tiled floor where the number of blue tiles $B$ used in the $n^{th}$ row is given by $B = 3n - 2$. In which row will the architect use exactly 145 blue tiles?

Question 4

4. A pattern of matchsticks forms a series of connected pentagons. 1 pentagon needs 5 sticks, 2 connected pentagons need 9 sticks, and 3 connected pentagons need 13 sticks. What is the general algebraic rule for $n$ connected pentagons?

A) $5n$

B) $n + 4$

C) $4n + 4$

D) $4n + 1$

Answer

Correct Answer: D

Solution: Let's analyze the difference progression pattern: * For $n = 1$: $5 = 4(1) + 1$ * For $n = 2$: $9 = 4(2) + 1$ * For $n = 3$: $13 = 4(3) + 1$ The common difference between terms is $+4$. The general rule for $n$ pentagons is $4n + 1$. Each additional pentagon attached requires 4 extra sticks because it shares exactly one boundary wall side with the previous one.

Question 5

5. Examine the structural number pattern: $1 \times 8 + 1 = 9$ $12 \times 8 + 2 = 98$ $123 \times 8 + 3 = 987$ Based on this progression, what is the value result of $123456 \times 8 + 6$?

A) 98765

B) 987654

C) 987645

D) 9876543

Answer

Correct Answer: B

Solution: The structural logic shows that the output yields a clean sequence of descending integer digits beginning from 9. The total number of digits contained within the answer directly matches the scalar unit added at the tail end. For $123456 \times 8 + 6$, the output must consist of 6 descending digits: **987654**.

Question 6

6. A sequence is defined by the rule: "Square the position number $n$ and subtract the previous position number $(n-1)$." What is the $8^{th}$ term of this sequence? (For $n=1$, assume the previous position is 0).

A) 64

B) 55

C) 57

D) 49

Answer

Correct Answer: C

Solution: The algorithmic pattern rule converts to the expression: $T_n = n^2 - (n-1)$. Evaluating for target position $n = 8$: $$T_8 = 8^2 - (8-1)$$ $$T_8 = 64 - 7 = 57$$

Question 7

7. The sum of the first $n$ consecutive odd numbers is always $n^2$. What is the sum of all odd numbers situated between 20 and 40?

A) 300

B) 400

C) 100

D) 210

Answer

Correct Answer: A

Solution: Sum of odd numbers between 20 and 40 can be calculated as: $$\text{Sum} = (\text{Sum of first 20 odd numbers up to 40}) - (\text{Sum of first 10 odd numbers up to 20})$$ * Odd numbers sequence up to 40 is $1, 3, \dots, 39$ (giving count $n = 20$). * Odd numbers sequence up to 20 is $1, 3, \dots, 19$ (giving count $n = 10$). $$\text{Target Sum} = 20^2 - 10^2 = 400 - 100 = 300.$$

Question 8

8. A pattern of spatial dots forms a classic "Pentagonal Number". The formula for the $n^{th}$ pentagonal number is given by $P_n = \frac{3n^2 - n}{2}$. What is the $4^{th}$ pentagonal number?

A) 12

B) 15

C) 26

D) 22

Answer

Correct Answer: D

Solution: Substitute the structural position $n = 4$ into the given formula: $$P_4 = \frac{3(4^2) - 4}{2}$$ $$P_4 = \frac{3(16) - 4}{2}$$ $$P_4 = \frac{48 - 4}{2} = \frac{44}{2} = 22.$$

Question 9

9. If the structural pattern layout of a variable sequence is $x, 2x+1, 3x+2, 4x+3, \dots$, what is the expression for the $10^{th}$ term in terms of $x$?

A) $10x + 10$

B) $10x + 9$

C) $9x + 10$

D) $11x + 10$

Answer

Correct Answer: B

Solution: Let's isolate and evaluate the linear coefficients pattern matching: * Term 1: $1x + 0$ * Term 2: $2x + 1$ * Term 3: $3x + 2$ * Term $n$: $nx + (n-1)$ Evaluating for sequence position $n = 10$ yields: $10x + (10-1) = 10x + 9$.

Question 10

10. A logical sequence is given as $2, 6, 12, 20, 30, \dots$. What is the value of the $20^{th}$ term?

A) 400

B) 410

C) 420

D) 380

Answer

Correct Answer: C

Solution: Let's check the product factor structure of the terms: * 1st term: $1 \times 2 = 2$ * 2nd term: $2 \times 3 = 6$ * 3rd term: $3 \times 4 = 12$ * 4th term: $4 \times 5 = 20$ The sequence defines rectangular numbers matching the general algebraic rule $n(n+1)$. Evaluating for target position $n = 20$: $20 \times 21 = 420$.

Accepted Answer

Correct Answer: A

Solution: We are given the target value $B = 145$. Set up and solve the linear equation for $n$: $$3n - 2 = 145$$ $$3n = 145 + 2$$ $$3n = 147$$ $$n = \frac{147}{3} = 49$$ Therefore, exactly 145 blue tiles are laid in the **$49^{th}$ row**.

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 5) | Concept of Maths

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 5) | Concept of Maths

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4. A pattern of matchsticks forms a series of connected pentagons. 1 pentagon needs 5 sticks, 2 connected pentagons need 9 sticks, and 3 connected pentagons need 13 sticks. What is the general algebraic rule for $n$ connected pentagons?

5. Examine the structural number pattern: <br> $1 \times 8 + 1 = 9$<br> $12 \times 8 + 2 = 98$<br> $123 \times 8 + 3 = 987$<br> Based on this progression, what is the value result of $123456 \times 8 + 6$?

6. A sequence is defined by the rule: "Square the position number $n$ and subtract the previous position number $(n-1)$." What is the $8^{th}$ term of this sequence? (For $n=1$, assume the previous position is 0).

7. The sum of the first $n$ consecutive odd numbers is always $n^2$. What is the sum of all odd numbers situated between 20 and 40?

8. A pattern of spatial dots forms a classic "Pentagonal Number". The formula for the $n^{th}$ pentagonal number is given by $P_n = \frac{3n^2 - n}{2}$. What is the $4^{th}$ pentagonal number?

9. If the structural pattern layout of a variable sequence is $x, 2x+1, 3x+2, 4x+3, \dots$, what is the expression for the $10^{th}$ term in terms of $x$?

10. A logical sequence is given as $2, 6, 12, 20, 30, \dots$. What is the value of the $20^{th}$ term?

Grade 6 Patterns in Mathematics Quiz (Hard) - Set 5 | Concept of Maths

1. A galactic explorer observes a pattern of stars forming a sequence. The number of stars in the $n^{th}$ formation follows the rule $S = n^2 + n + 1$. How many stars will be in the $12^{th}$ formation?

2. Consider a pattern where each term is the sum of the preceding two terms (starting with 1 and 2). If the sequence is $1, 2, 3, 5, 8, 13, \dots$, what is the $10^{th}$ term?

3. An architect designs a tiled floor where the number of blue tiles $B$ used in the $n^{th}$ row is given by $B = 3n - 2$. In which row will the architect use exactly 145 blue tiles?

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 5) | Concept of Maths

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 5) | Concept of Maths

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4. A pattern of matchsticks forms a series of connected pentagons. 1 pentagon needs 5 sticks, 2 connected pentagons need 9 sticks, and 3 connected pentagons need 13 sticks. What is the general algebraic rule for $n$ connected pentagons?

5. Examine the structural number pattern: <br> $1 \times 8 + 1 = 9$<br> $12 \times 8 + 2 = 98$<br> $123 \times 8 + 3 = 987$<br> Based on this progression, what is the value result of $123456 \times 8 + 6$?

6. A sequence is defined by the rule: "Square the position number $n$ and subtract the previous position number $(n-1)$." What is the $8^{th}$ term of this sequence? (For $n=1$, assume the previous position is 0).

7. The sum of the first $n$ consecutive odd numbers is always $n^2$. What is the sum of all odd numbers situated between 20 and 40?

8. A pattern of spatial dots forms a classic "Pentagonal Number". The formula for the $n^{th}$ pentagonal number is given by $P_n = \frac{3n^2 - n}{2}$. What is the $4^{th}$ pentagonal number?

9. If the structural pattern layout of a variable sequence is $x, 2x+1, 3x+2, 4x+3, \dots$, what is the expression for the $10^{th}$ term in terms of $x$?

10. A logical sequence is given as $2, 6, 12, 20, 30, \dots$. What is the value of the $20^{th}$ term?