angepepagiortzb

2021-11-15

Use the summation formulas to rewrite the expression withous summatiom notation.
$\sum _{i=1}^{n}\frac{4i+5}{{n}^{2}}$
Use the result to find the sums for n=10, 100.

Mollicchiuk

$\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}$
$\sum _{i=1}^{n}k=n.k$, where k=constant
for $n=10$
$\sum _{i=1}^{10}\frac{4i+5}{{\left(10\right)}^{2}}=\frac{1}{100}\left[4\sum _{i=1}^{10}i+\sum _{i=1}^{10}5\right]$
$=\frac{1}{100}\left[\right]4\frac{10\left(10+1\right)}{2}+10×5\right]=2.7$
for n=100
$\sum _{i=1}^{100}\frac{4i+5}{{\left(100\right)}^{2}}=\frac{1}{{\left(100\right)}^{2}}\left[4\cdot \sum _{i=1}^{100}i+\sum {\left\{i=1\right\}}^{100}5\right]$
$=\frac{1}{{\left(100\right)}^{2}}\left[4\frac{100\left(100+1\right)}{2}+\left(5×100\right)\right]$
$=2.07$

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