Answer
250
Work Step by Step
$S = \sum_{k=1}^{25}\left(\frac{3 x+1}{4}\right)$ (Given the sum. Note that $a_{n} = \frac{3n+1}{4}$. Hence this is an arithmetic sequence since difference of consecutive two terms is constant.)
$n=25$
$a_{1}=\frac{3 \cdot 1+1}{4}=\frac{4}{4}=1$ (Compute first term)
$a_{25}=\frac{3 \cdot 25+1}{4}=\frac{76}{4}=19$ (Compute last term for n = 25)
$S_{25}=\frac{25(1+19)}{2}=\frac{500}{2}=250$ (we compute the sum $S$ using the formula for arithmetic sequence given by $S_{n} = \frac{n(a_{n} + a_{1})}{2}$)
