Question

Sum $3,-4,\frac{16}{3},....$ to $2n$ terms.

Answer 1

Clearly the given series is G.P. with common ratio $r=\frac{-4}{3}$
$S_n$ be the sum of first $n$ terms of G.P
Formula for sum of $n$ terms of G.P. is given by
$S_n=\frac{a(r^n-1)}{r-1}$
$n=2n$ ........ (given)

$S_{2n}=\frac{3\times ({\left(\frac{-4}{3}\right)}^{2n}-1)}{\frac{-4}{3}-1}$

$S_{2n}=\frac{9\times ({\left(\frac{16}{9}\right)}^n-1)}{-7}$

$\Rightarrow S_{2n}=\frac{9}{7}\{1-{\left(\frac{4}{3}\right)}^{2n}\}$.