Question

Calculate the sum of $3,\sqrt{3},1,....$

Answer 1

Number of terms $n=\infty$
Clearly, given series is G.P,
Common ratio $r=\frac{1}{\sqrt{3}}$
First term $a=3$
Formula for sum of $n$ terms of G.P. is given by
$S_n=\frac{a(r^n-1)}{r-1}$
if $n=\infty$ & $|r|<1$ then formula for total sum is
$S=\frac{a}{1-r}$
Substituting the values, we get
$S=\frac{3}{1-\frac{1}{\sqrt{3}}}$
$\Rightarrow S=\frac{3\sqrt{3}}{\sqrt{3}-1}$

$\Rightarrow S=\frac{3\sqrt{3}}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}$

$\Rightarrow S=\frac{3(3+\sqrt{3})}{2}$