Question

The sum of $5^{th}$ and $9^{th}$ terms of an A.P. is 30. If its ${25}^{th}$ term is three times its $8^{th}$ term, find the AP.

Answer 1

nth term $t_n=a+(n-1)d$

$a=$ First term

$d=$ Common difference

$n=$ number of terms

$t_n=n^{th}$ term

Given

$T_5+T_9=30$

$a+4d+a+8d=30$

$2a+12d=30$

$a+6d=15$

$a=-6d+15$

now

$T_{25}=3T_8$

$a+24d=3(a+7d)$

$24d-21d=3a-a$

$3d=2a$

$3d=2(6d-15)$

$15d=30$

$d=2$

$a=15-6\times 2$

$a=3$

The AP is $3,5,7,\mathrm{9.........}$