Question

The sum of the first 10 terms of an arithmetic progression is 150. If the first term is 10, what is the common difference?

• A. 10/5
• B. 10/3
• C. 10/9
• D. 10/2

Hint:

Remember: The sum of an arithmetic progression can be calculated using the formula \( S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \).

Answer 1

The Correct Option is C

Step 1: Comprehend the Problem and Recall the Formula

An arithmetic progression (AP) features a constant increment between consecutive terms, termed the common difference (d). We are provided with:

• Number of terms, n = 10
• First term, a = 10
• Sum of the initial 10 terms, S₁₀ = 150

The objective is to determine the common difference, d. The formula for the sum of the first n terms of an AP is:

Sₙ = (n/2) × [2a + (n - 1)d]

Step 2: Input Given Values into the Formula

Substitute n = 10, a = 10, and S₁₀ = 150 into the sum formula:

150 = (10/2) × [2(10) + (10 - 1)d]

Simplify the expression:

150 = 5 × [20 + 9d]

Step 3: Isolate the Expression Within the Brackets

Divide both sides by 5 to simplify:

150 / 5 = 20 + 9d

30 = 20 + 9d

Subtract 20 from both sides:

30 - 20 = 9d

10 = 9d

Step 4: Calculate the Common Difference

Solve for d by dividing both sides by 9:

d = 10 / 9

The common difference is 10/9, approximately 1.111.

Step 5: Validate the Solution

Verify by computing the sum with d = 10/9. The first term is 10. The 10th term of the AP is found using aₙ = a + (n - 1)d.

For the 10th term (n = 10):

a₁₀ = 10 + (10 - 1)(10/9)

a₁₀ = 10 + 9(10/9)

a₁₀ = 10 + 10 = 20

The sum can also be calculated using Sₙ = (n/2) × (first term + last term):

S₁₀ = (10/2) × (10 + 20)

S₁₀ = 5 × 30 = 150

The calculated sum matches the given value of 150, confirming the correctness of the common difference!

Final Answer: Option (3)

The common difference of the arithmetic progression is 10/9 (approximately 1.111).