Question

If the sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = 5n - n^2 \), then the common difference of the sequence is:

• A. \( 4 \)
• B. \( -2 \)
• C. \( 2 \)
• D. \( 6 \)

Hint:

To find the common difference from the sum formula, use: \[ d = a_2 - a_1. \]

Answer 1

The Correct Option is B

We know that:
\[ a_n = S_n - S_{n-1}. \] First term:
\[ a_1 = S_1 = 5(1) - (1)^2 = 5 - 1 = 4. \] Second term:
\[ a_2 = S_2 - S_1. \] \[ S_2 = 5(2) - (2)^2 = 10 - 4 = 6. \] \[ a_2 = 6 - 4 = 2. \] Common difference:
\[ d = a_2 - a_1 = 2 - 4 = -2. \]