Question

In an AP the first term is 2 and sum of first five term is one-fourth of the next five terms . show that 20th term is -112.

Answer 1

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Assumption

${\boxed{\sf\:{First\;term\;be\;a}}}$

Also

${\boxed{\sf\:{Common\;difference\;be\;d}}}$

Hence,

AP will be :-

2 , 2 + d , 2 + 2d , 2 + 3d .......

Now

${\boxed{\sf\:{Sum\;of\;first\;five\;terms}}}$

= 10 + 10d

${\boxed{\sf\:{Sum\;of\;next\;five\;terms}}}$

= 10 + 35d

Situation

$\tt{\rightarrow 10+10d=\dfrac{1}{4}(10+35d)}$

40 + 40d = 10 + 35d

40d - 35d = 10 - 40

5d = -30

$\tt{\rightarrow d=\dfrac{-30}{5}}$

d = -6

Now,

$\tt{\rightarrow a_{n}=a+(n-1)d}$

$\tt{\rightarrow a_{20}=a+(20-1)d}$

$\tt{\rightarrow a_{20}=2+(19)(-6)}$

$\tt{\rightarrow a_{20}=2-114}$

$\tt{\rightarrow a_{20}=-112}$

$\Large{\boxed{\sf\:{Hence\;Proved}}}$