Question

The sum of first $10$ term of an A.P is one third of sum of next $10$ term. If first term is $-5$.. Find the sum of its first $30$ terms

Answer 1

Given,

$a=-5$

$S_{10}=5[-10+9d]$

$S_{20}=10[-10+19d]$

$S_{10}=\frac{1}{3}(S_{20}-S_{10})$

$5[-10+9d]=\frac{1}{3}\left\{10\right[-10+19d]-5[-10+9d\left]\right\}$

$15[-10+9d]=\left\{10\right[-10+19d]-5[-10+9d\left]\right\}$

$-150+135d=\{-100+190d+50-45d\}$

$-150+135d=-50+145d$

$-100=10d$

$\therefore d=-10$

$S_{30}=\frac{30}{2}\left[2\right(-5)+(30-1\left)\right(-10\left)\right]$

$=15[-10+29(-10\left)\right]$

$=-4500$