Question

If the sum of the first five terms of an $AP$ is equal to one-fourth of the sum of next five terms of the $AP$ and the first term of the $AP$ is $2,$ then find the sum of the first $20$ terms of the $AP.$

Answer 1

Let a be the First term of the AP.

a = 2

sum of first 10 terms $S_{10}=\frac{10}{2}[2a+(10-1\left)d\right]$

$S_{10}=20+45d$ ---(1) $\because sn=\frac{n}{2}\left[2a\right(n-1\left)d\right]$

sum of first 5 terms $S_5\frac{5}{2}[2\times 2+(5-1\left)d\right]$

$S_5=10+10d------\left(2\right)$

for calculation of next first term is

$S_{10}-S_5=10+35d$

sum of first five terms = $\frac{1}{4}$ sum of next 5

$10+10d=\frac{1}{4}[10+35d]$

40+40d=10+35d

d = -6

$S_{20}=\frac{20}{2}[2\times 2+(20-1\left)\right(-6\left)\right]$

$=10[4-19\times 6]$

= 10[4-114] = -1100