Answer
$S_{25}=3,775$
Work Step by Step
RECALL:
(1) The sum of the first $n$ terms, $s_n$, of an arithmetic sequence can be found using the formula
$S_n = \frac{n}{2}(a_1+a_n)$
where
$a_1$=first term
$a_n$ = $n^{th}$ term
(2) The $n^{th}$ term, $a_n$, of an arithmetic sequence can be found using the formula
$a_n=a_1 + d(n-1)$
where
$d$=common difference
$a_1$ = first term
To find the sum of the first 25 terms of the sequence, we need to find the value of $a_{25}$. However, the value of $a_{25}$ can only be found if we know the value of $d$.
The terms of the sequence increase by 12 so the common difference is $d=12$.
The first term of the sequence is $7$ so $a_1=7$.
Substitute these values into the formula in (2) above to obtain;
$a_n= 7+12(n-1)$
Solve for the 25th term of the sequence to obtain:
$a_{25} = 7 + 12(25-1)
\\a_{25} = 7 + 12(24)
\\a_{25} = 7 + 288
\\a_{25} = 295$
Solve for the sum of the first 25 terms using the formula in (1) above to obtain:
$S_{25} = \frac{25}{2}(7+295)
\\S_{25} = \frac{25}{2}(302)
\\S_{25} = 3775$
