Question

Find the sum of the first 22 terms of an A.P. in which the common difference is 7 and the 22nd term is 149.

Hint:

For sum of an arithmetic sequence, use: \[ S_n = \frac{n}{2} [2a + (n-1)d]. \]

Answer 1

We use the formula for the \( n \)th term of an arithmetic progression:
\[ a_n = a + (n - 1)d. \] Given:
\[ a_{22} = 149, \quad d = 7, \quad n = 22. \] Substituting the values:
\[ 149 = a + (22 - 1) \times 7. \] \[ 149 = a + 21 \times 7. \] \[ 149 = a + 147. \] \[ a = 2. \] Now, we calculate the sum of the first 22 terms:
\[ S_n = \frac{n}{2} [2a + (n-1)d]. \] \[ S_{22} = \frac{22}{2} [2(2) + (22-1) \times 7]. \] \[ = 11 [4 + 21 \times 7]. \] \[ = 11 [4 + 147]. \] \[ = 11 \times 151. \] \[ = 1661. \]