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 ${20}^{th}$ term is $-112$.

Answer 1

First term $=2.$
Let $d$ be the common difference of the A.P.
Therefore, the A.P. is $2,2+d,2+2d,2+3d,...$
Sum of first five terms $=10+10d$
Sum of next five terms $=10+35d$
According to the given condition ,
$10+10d=\frac{1}{4}(10+35d)\Rightarrow 40+40d=10+35d\Rightarrow 30=-5d\Rightarrow d=-6\therefore a_{20}=a+(20-1)d=2+\left(19\right)(-6)=2-114=-112$
Thus, the ${20}^{th}$ term of the A.P. is $-112.$