Answer
$S(n)=\displaystyle \frac{7n+15}{2n}$
$S(10)=\displaystyle \frac{17}{4}=4.25$
$S(100)=3.575$
$S(1000)=3.5075$
$S(10,000)=3.50075$
Work Step by Step
$S(n)= \displaystyle \sum_{j=1}^{n}\frac{7j+4}{n^{2}}=$
... the $n^{2}$ is constant,...$\displaystyle \sum_{i=1}^{n}ka_{i}=k\sum_{i=1}^{n}a_{i}$
=$\displaystyle \frac{1}{n^{2}}\sum_{j=\mathrm{I}}^{n}(7j+4)$
... property: $ \displaystyle \sum_{i=1}^{n}(a_{i}\pm b_{i})=\sum_{i=1}^{n}a_{i}\pm\sum_{i=1}^{n}b_{i}$
=$\displaystyle \frac{1}{n^{2}}[\sum_{j=\mathrm{I}}^{n}(7j)+\sum_{j=\mathrm{I}}^{n}4]$
... Th4.2.2. $\displaystyle \sum_{i=1}^{n}i=\frac{n(n+1)}{2}$, Th4.2.1. $\displaystyle \sum_{i=1}^{n}c=cn$
$=\displaystyle \frac{1}{n^{2}}[7\cdot\frac{n(n+1)}{2}+4n]$
$=\displaystyle \frac{7n^{2}+7n}{2n^{2}}+\frac{4n}{n^{2}}=\frac{n(7n+7)}{2n^{2}}+\frac{4}{n}$
$=\displaystyle \frac{7n+7+8}{2n}$
$S(n)=\displaystyle \frac{7n+15}{2n}$
$S(10)=\displaystyle \frac{70+15}{20}=4.25$
$S(100)=\displaystyle \frac{700+15}{200}=3.575$
$S(1000)=\displaystyle \frac{7000+15}{2000}=3.5075$
$S(10,000)=\displaystyle \frac{70,000+15}{20,000}=3.50075$
