Question

The sum of the first 22 terms of the AP \( 8, 3, -2, \dots \) is

• A. \( -979 \)
• B. \( 979 \)
• C. \( 1028 \)
• D. \( -1028 \)

Hint:

The sum of the first \( n \) terms of an AP is \( S_n = \frac{n}{2} [2a_1 + (n - 1)d] \) or \( S_n = \frac{n}{2} [a_1 + a_n] \), where \( a_n \) is the \( n^{th} \) term.

Answer 1

The Correct Option is A

Step 1: Identify the first term and the common difference.
In the given arithmetic progression \( 8, 3, -2, \dots \), the first term is \( a_1 = 8 \).
The common difference \( d \) is the difference between consecutive terms: \[ d = a_2 - a_1 = 3 - 8 = -5 \] We can verify this: \[ d = a_3 - a_2 = -2 - 3 = -5 \] Step 2: Use the formula for the sum of the first \( n \) terms of an AP.
The sum of the first \( n \) terms of an arithmetic progression is given by the formula: \[ S_n = \frac{n}{2} [2a_1 + (n - 1)d] \] We need to find the sum of the first 22 terms, so \( n = 22 \). Step 3: Substitute the values into the formula. \[ S_{22} = \frac{22}{2} [2(8) + (22 - 1)(-5)] \] \[ S_{22} = 11 [16 + (21)(-5)] \] \[ S_{22} = 11 [16 - 105] \] \[ S_{22} = 11 [-89] \] \[ S_{22} = -979 \] The sum of the first 22 terms of the arithmetic progression is \( -979 \).