Question 1

1. A gardener is planting marigolds in a triangular arrangement. The first row has 3 plants, the second has 7 plants, and the third has 11 plants. If this arithmetic pattern continues, what is the rule for the number of plants in the n-th row?

A) 3n + 4B) 4n - 1C) 4n + 3D) 3n - 1

Answer

✓ Correct Answer: Option B

Explanation: Solution: Row 1 (n=1): 3 plants. Row 2 (n=2): 7 plants. Row 3 (n=3): 11 plants. The difference between consecutive rows is 7 - 3 = 4. The rule follows the form 4n + c. For n=1, 4(1) + c = 3 \Rightarrow c = -1. Thus, the rule is 4n - 1.

Question 2

2. To make a chain of x triangles using matchsticks, where each triangle shares one side with the next, the number of matchsticks required follows a pattern. 1 triangle needs 3 matches, 2 triangles need 5 matches, and 3 triangles need 7 matches. What is the expression for x triangles?

2x + 1

Question 3

3. A local taxi service in Ranchi charges a fixed amount of ₹500 as a booking fee and an additional ₹15 for every kilometer (d) traveled. Which expression represents the total cost C?

A) C = 500d + 15B) C = 515dC) C = 15d + 500D) C = 15(d + 500)

Answer

✓ Correct Answer: Option C

Explanation: Solution: The cost consists of a constant (fixed) part and a variable part depending on distance. Fixed cost = ₹500. Variable cost = $15 \times d$. Total Cost C = 15d + 500.

Question 4

4. Isha is designing a walkway with square tiles. Section 1 uses 5 tiles, Section 2 uses 9 tiles, and Section 3 uses 13 tiles. Which expression describes the number of tiles in Section n?

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

Answer

✓ Correct Answer: Option D

Explanation: Solution: The sequence is 5, 9, 13.... The common difference is 4. The rule is 4n + c. Since 4(1) + c = 5, then c = 1. The rule is 4n + 1.

Question 5

5. A software engineer has a starting salary of S. Every year, she receives a fixed increment of I. What will be her salary after y years?

A) S + IB) $S + (y \times I)$C) y(S + I)D) S - yI

Answer

✓ Correct Answer: Option B

Explanation: Solution: After 1 year, salary = S + I. After 2 years, salary = S + 2I. After y years, the total increment is $y \times I$. So, salary = S + yI.

Question 6

6. Consider a row of k identical square tables pushed together. One table seats 4 people. Two tables seat 6 people. Three tables seat 8 people. How many people can be seated around k tables?

A) 2k + 2B) 4kC) 3k + 1D) 2k + 4

Answer

✓ Correct Answer: Option A

Explanation: Solution: Sequence: 4, 6, 8.... The difference is 2. The formula is 2k + c. For k=1, 2(1) + c = 4 \Rightarrow c = 2. The rule is 2k + 2. Alternatively, 2 people sit at the ends, and 2 people sit at each table along the sides ($2 \times k$). Total = 2k + 2.

Question 7

7. Observations show that the sum of the first n odd natural numbers follows a specific pattern: $1 = 1^2$, $1+3 = 4 = 2^2$, $1+3+5 = 9 = 3^2$. What is the sum of the first n odd numbers?

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

Answer

✓ Correct Answer: Option C

Explanation: Solution: The pattern clearly shows that the sum of the first n odd numbers is the square of n. Sum = $n \times n = n^2$.

Question 8

8. A baker prepares b batches of cookies. Each batch contains 12 cookies. If the baker eats 5 cookies from the total amount, which expression represents the cookies remaining?

A) 12 + b - 5B) 5b - 12C) 12(b - 5)D) 12b - 5

Answer

✓ Correct Answer: Option D

Explanation: Solution: Total cookies produced = $12 \times b$ (12 cookies per batch). Cookies remaining after eating 5 = Total - 5 = 12b - 5.

Question 9

9. A wooden ladder has rungs spaced 25 cm apart. If there are r rungs, the distance from the first rung to the r-th rung is given by which pattern?

A) 25rB) 25(r - 1)C) 25r + 25D) r + 25

Answer

✓ Correct Answer: Option B

Explanation: Solution: The number of gaps between r rungs is (r-1). Since each gap is 25 cm, the total distance is $25 \times (r-1)$.

Question 10

10. Aarav saves money every week. In week 1 he saves ₹50. In week 2 he saves ₹80. In week 3 he saves ₹110. If the pattern continues, how much does he save in week w?

A) 30w + 20B) 30w + 50C) 50w + 30D) 80w - 30

Answer

✓ Correct Answer: Option A

Explanation: Solution: The savings are 50, 80, 110.... The common difference is 30. Rule: 30w + c. For w=1, 30(1) + c = 50 \Rightarrow c = 20. The expression is 30w + 20.

Accepted Answer

✓ Correct Answer: Option A

Explanation: Solution: Let's list the matches: x=1 \to 3, x=2 \to 5, x=3 \to 7. This is a sequence increasing by 2. The formula is 2x + 1. For x=1, 2(1)+1=3. For x=2, 2(2)+1=5.

Grade 6 : Mathematics (Ganita Prakash) - Patterns in Mathematics (Set 2)

1. A gardener is planting marigolds in a triangular arrangement. The first row has 3 plants, the second has 7 plants, and the third has 11 plants. If this arithmetic pattern continues, what is the rule for the number of plants in the n-th row?

2. To make a chain of x triangles using matchsticks, where each triangle shares one side with the next, the number of matchsticks required follows a pattern. 1 triangle needs 3 matches, 2 triangles need 5 matches, and 3 triangles need 7 matches. What is the expression for x triangles?

3. A local taxi service in Ranchi charges a fixed amount of ₹500 as a booking fee and an additional ₹15 for every kilometer (d) traveled. Which expression represents the total cost C?

4. Isha is designing a walkway with square tiles. Section 1 uses 5 tiles, Section 2 uses 9 tiles, and Section 3 uses 13 tiles. Which expression describes the number of tiles in Section n?

5. A software engineer has a starting salary of S. Every year, she receives a fixed increment of I. What will be her salary after y years?

6. Consider a row of k identical square tables pushed together. One table seats 4 people. Two tables seat 6 people. Three tables seat 8 people. How many people can be seated around k tables?

7. Observations show that the sum of the first n odd natural numbers follows a specific pattern: $1 = 1^2$, $1+3 = 4 = 2^2$, $1+3+5 = 9 = 3^2$. What is the sum of the first n odd numbers?

8. A baker prepares b batches of cookies. Each batch contains 12 cookies. If the baker eats 5 cookies from the total amount, which expression represents the cookies remaining?

9. A wooden ladder has rungs spaced 25 cm apart. If there are r rungs, the distance from the first rung to the r-th rung is given by which pattern?

10. Aarav saves money every week. In week 1 he saves ₹50. In week 2 he saves ₹80. In week 3 he saves ₹110. If the pattern continues, how much does he save in week w?