# Use the summation formulas to rewrite the expression without the

Use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for $n=10,100,1000,$ and
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rodclassique4r
$\sum _{i=1}^{n}\frac{2i+1}{{n}^{2}}$
$=\frac{1}{{n}^{2}}\sum _{i=1}^{n}2i+1$
$=\frac{2}{{n}^{2}}\sum _{i=1}^{n}i+\frac{1}{{n}^{2}}\sum _{i=1}^{n}1$
Remember that:
$\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}$
and
$\sum _{i=1}^{n}1=n$
$=\frac{2}{{n}^{2}}\frac{n\left(n+1\right)}{2}+\frac{1}{{n}^{2}}\cdot n$
$=\frac{n+1}{n}+\frac{1}{n}$
$=\frac{n+2}{n}$
$=1+\frac{2}{n}$
$\overline{)\begin{array}{cc}n& 1+\frac{2}{n}\\ 10& 1.2\\ 100& 1.02\\ 1000& 1.002\\ 10000& 1.0002\end{array}}$

Joseph Fair