Question

How do you find the sum of the first 25 terms of an arithmetic sequence whose 7th term is −247 and whose 18th term is −49?

Answer 1

The sum of the first 25 terms is `color(red)(-3475)`.

Explanation:

We know that `a_7 = -247` and `a_18 = -49`.

We also know that `a_n = a_1 + (n-1)d`.

`a_18 = a_1 + (18-1)d`

Equation (1): `-49 = a_1 + 17d`

and

`a_7 = a_1 + (7-1)d`

Equation (2):`-247 = a_1 + 6d`

Subtract Equation (2) from Equation (1).

`-49 + 247 = 17d -6d`

`198 = 11d`

Equation (3): `d= 18`

Substitute Equation (3) in Equation (1).

`-49 = a_1 + 17d`

`-49 = a_1 + 17×18 = a_1 + 306`

`a_1 = -49-306 = -355`

So `a_1 = -355` and `d = 18`.

`a_n = a_1 + (n-1)d`

So the 25th term is given by

`a_25 = -355 + (25-1)×18 = -355 + 24×18 = -355 + 432`

`a_25 = 77`

The sum `S_n` of the first `n` terms of an arithmetic series is given by

`S_n = (n(a_1+a_n))/2`

So

`S_25 = (25(a_1+a_25))/2 = (25(-355+77))/2 = (25(-278))/2`

`S_25 = -3475`