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?

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Answer 1 · 2015-07-17T21:13:24

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

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

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

$S_25 = -3475$