Evaluate the following: _ n=2 ^ 10 (4^n) - Vidyadip

Question

Evaluate the following:$\sum_\limits{\text{n}=2}^{10}(4^\text{n})$

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Answer

$\sum_\limits{\text{n}=2}^{10}(4^\text{n})$
$=4^2+4^3+4^4+\ \dots\ +4^{10}$
$\text{a}=4^2,\text{r}=\frac{4^3}{4}=4,\text{n}=9$
$\text{S}_{10}=\frac{\text{a}(\text{r}^9-1)}{1-\text{r}}$
$=\frac{4^2(4^9-1)}{4-1}$
$=\frac13\big[4^{11}-16\big]$
$=\frac{16}{3}\big[4^{9}-1\big]$

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