Question

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Answer 1

Given:
$a_{24}=2a_{10}\Rightarrow a+23d=2(a+9d)\Rightarrow a+23d=2a+18d$
$\Rightarrow a=5d\dots \dots \left(i\right)$
$a_{72}=a+(72-1)d$
$a_{72}=a+71d$ $[\because a=5d\text{from (i)}]$
$\Rightarrow a_{72}=76d\dots \dots \left(ii\right)$
$a_{34}=a+(34-1)d$
$=5d+33d$ $[\because a=5d\text{from (i)}]$
$=38d\dots \dots \left(iii\right)$
From (ii) and (iii)
$a_{72}=2a_{34}$
Hence, proved.