Find the sum of the GP :


Find the sum of the GP :

$\sqrt{2}+\frac{1}{\sqrt{2}}+\frac{1}{2 \sqrt{2}}+\ldots \ldots$ to 8 terms



Sum of a G.P. series is represented by the formula $\mathrm{S}_{\mathrm{n}}=\mathrm{a} \frac{1-\mathrm{r}^{\mathrm{n}}}{1-\mathrm{r}}$

hen |r|<1. ‘Sn’ represents the sum of the G.P. series upto nth terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.



$r=($ ratio between the $n$ term and $n-1$ term $) \frac{1}{\sqrt{2}} \div \sqrt{2}=\frac{1}{2}$

n = 8 terms

$\therefore \mathrm{S}_{\mathrm{n}}=\sqrt{2} \times \frac{1-\frac{1}{2}^{8}}{1-\frac{1}{2}}$

$\Rightarrow \mathrm{S}_{\mathrm{n}}=\sqrt{2} \times \frac{1-\frac{1}{256}}{\frac{1}{2}}$

$\Rightarrow \mathrm{S}_{\mathrm{n}}=\sqrt{2} \times \frac{\frac{255}{256}}{\frac{1}{2}}$

$\Rightarrow \mathrm{S}_{\mathrm{n}}=\sqrt{2} \times \frac{255}{128}$

$\therefore S_{\mathrm{n}}=\frac{255 \sqrt{2}}{128}$


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