Question

The sum of the first 30 terms of an arithmetic progression is 930. If the first term is 2, what is the common difference of the progression?

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Use the formula for the sum of an arithmetic progression to solve for the common difference when the sum and first term are known. Ensure the correct substitution of values in the equation.

Answer 1

The Correct Option is B

We are given that the sum of the first 30 terms of an arithmetic progression (AP) is 930, and the first term is 2. We need to find the common difference \( d \). The formula for the sum of the first \( n \) terms of an arithmetic progression is: \[ S_n = \frac{n}{2} \left[ 2a + (n-1) \cdot d \right] \] 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. Substituting the known values: \[ S_{30} = \frac{30}{2} \left[ 2(2) + (30-1) \cdot d \right] \] \[ 930 = 15 \left[ 4 + 29d \right] \] \[ 930 = 15(4 + 29d) \] \[ 930 = 60 + 435d \] \[ 930 - 60 = 435d \] \[ 870 = 435d \] \[ d = \frac{870}{435} \] \[ d = 2 \] Thus, the common difference is \( d = 2 \).