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 starsB) 156 starsC) 157 starsD) 169 stars

Answer

✓ Correct Answer: Option C

Explanation: 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?

55

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?

A) 49th rowB) 48th rowC) 50th rowD) 145th row

Answer

✓ Correct Answer: Option A

Explanation: 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**.

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) 5nB) n + 4C) 4n + 4D) 4n + 1

Answer

✓ Correct Answer: Option D

Explanation: 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) 98765B) 987654C) 987645D) 9876543

Answer

✓ Correct Answer: Option B

Explanation: 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) 64B) 55C) 57D) 49

Answer

✓ Correct Answer: Option C

Explanation: 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) 300B) 400C) 100D) 210

Answer

✓ Correct Answer: Option A

Explanation: Solution: Sum of odd numbers between 20 and 40 can be calculated as: $Sum = (Sum of first 20 odd numbers up to 40) - (Sum of first 10 odd numbers up to 20)$ * 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). $$Target Sum = 20^2 - 10^2 = 400 - 100 = 300.$$

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) 12B) 15C) 26D) 22

Answer

✓ Correct Answer: Option D

Explanation: 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 + 10B) 10x + 9C) 9x + 10D) 11x + 10

Answer

✓ Correct Answer: Option B

Explanation: 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) 400B) 410C) 420D) 380

Answer

✓ Correct Answer: Option C

Explanation: 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: Option B

Explanation: 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.

Grade 6 Patterns in Mathematics Quiz (Hard) - Set 5

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?

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?