Arithmetic Progression

Step 1: Understand the Problem and Recall the Formula

In an arithmetic progression (AP), each term increases by a constant value called the common difference, denoted as d. We’re given:

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

We need to find 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: Substitute the Given Values into the Formula

Using the sum formula, substitute n = 10, a = 10, and S₁₀ = 150:

150 = (10/2) × [2(10) + (10 - 1)d]

Simplify the expression:

150 = 5 × [20 + 9d]

Step 3: Solve for the Expression Inside 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: Solve for the Common Difference

Now, solve for d by dividing both sides by 9:

d = 10 / 9

So, the common difference is approximately 10/9 or about 1.111.

Step 5: Verify the Solution

Let’s verify by calculating the sum using d = 10/9. The first term is 10, and the 10th term of the AP can be found using the formula for the nth term: 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 the alternative sum formula: Sₙ = (n/2) × (first term + last term):

S₁₀ = (10/2) × (10 + 20)

S₁₀ = 5 × 30 = 150

The sum matches the given value of 150, so our common difference is correct!

Final Answer: Option (3)

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