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. 5
• B. 3
• C. 4
• D. 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: Use the formula for the sum of the first \( n \) terms of an arithmetic progression The sum \( S_n \) of the first \( n \) terms of an arithmetic progression is given by: \[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \] where: - \( S_n \) is the sum of the first \( n \) terms, - \( a \) is the first term, - \( d \) is the common difference, - \( n \) is the number of terms. Step 2: Substitute the known values We are given: - \( S_{10} = 150 \), - \( a = 10 \), - \( n = 10 \). Substitute these values into the sum formula: \[ 150 = \frac{10}{2} \left( 2 \times 10 + (10-1)d \right) \] \[ 150 = 5 \left( 20 + 9d \right) \] \[ 150 = 100 + 45d \] \[ 50 = 45d \] \[ d = \frac{50}{45} = \frac{10}{9} \approx 4 \] Answer: Therefore, the common difference is 4. So, the correct answer is option (3).