Question

In an A.P. the first term in 2 and the sum of the first five terms is one-fourth of the next five terms, Show that ${20}^{th}$ term is-112.

Answer 1

Here a = 2 and sum of first is -112

$\therefore \frac{1}{4}\times$ sum of next five terms .

$\therefore s_5=\frac{1}{4}\left(S_{10-S_5}\right)$

$\Rightarrow 4S_5=S_{10-S_5}$

$\Rightarrow 5S_5=S_{1}0$

$\Rightarrow 5\left[\frac{5}{2}\right\{2\times 2+(5-1)\left\}\right]=\frac{10}{2}[2\times 2+(10-1\left)d\right]$

$\Rightarrow \frac{25}{2}[4+4d]=5[4+9d]$

$\Rightarrow 25[4+4d]=10[4+4d]$

$\Rightarrow 100+100d=40+90d$

$\Rightarrow 100d-90d=40-100$

$\Rightarrow 10d=-60$

$\Rightarrow d=-6$

$Now,a_n=a+(n-1)d$

$\therefore a_{20}=2+(20-1)\times -6$

$\Rightarrow a_{20}=2-114=-112.$