# 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?

Jul 17, 2015

The sum of the first 25 terms is $\textcolor{red}{- 3475}$.

#### Explanation:

We know that ${a}_{7} = - 247$ and ${a}_{18} = - 49$.

We also know that ${a}_{n} = {a}_{1} + \left(n - 1\right) d$.

${a}_{18} = {a}_{1} + \left(18 - 1\right) d$

Equation (1): $- 49 = {a}_{1} + 17 d$

and

${a}_{7} = {a}_{1} + \left(7 - 1\right) d$

Equation (2):$- 247 = {a}_{1} + 6 d$

Subtract Equation (2) from Equation (1).

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

$198 = 11 d$

Equation (3): $d = 18$

Substitute Equation (3) in Equation (1).

$- 49 = {a}_{1} + 17 d$

-49 = a_1 + 17×18 = a_1 + 306

${a}_{1} = - 49 - 306 = - 355$

So ${a}_{1} = - 355$ and $d = 18$.

${a}_{n} = {a}_{1} + \left(n - 1\right) d$

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

${S}_{n} = \frac{n \left({a}_{1} + {a}_{n}\right)}{2}$

So

${S}_{25} = \frac{25 \left({a}_{1} + {a}_{25}\right)}{2} = \frac{25 \left(- 355 + 77\right)}{2} = \frac{25 \left(- 278\right)}{2}$

${S}_{25} = - 3475$