Question

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

Answer 1

Given:
$a=2,S_5=\frac{1}{4}(S_{10}-S_5)$
We have:
$S_5=\frac{5}{2}[2\times 2+(5-1\left)d\right]\Rightarrow S_5=5[2+2d]\dots \dots \left(i\right)$
Also, $S_{10}=\frac{10}{2}[2\times 2+(10-1\left)d\right]\Rightarrow S_{10}=5[4+9d]\dots \dots \left(ii\right)\because S_5=\frac{1}{4}(S_{10}-S_5)$
From (i) and (ii), we have:
$\Rightarrow 5[2+2d]=\frac{1}{4}\left[5\right(4+9d)-5(2+2d\left)\right]\Rightarrow 8+8d=4+9d-2-2d\Rightarrow d=-6\therefore a_{20}=a+(20-1)d\Rightarrow a_{20}=a+19d\Rightarrow a_{20}=2+19(-6)\Rightarrow a_{20}=-112$