Question

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

Answer 1

Given $t_5+t_9=30$.........(1)

and $t_{25}=3t_8$..............(2)

Solving equation (1),

$a+(5-1)d+a+(9-1)d=30$

$\therefore a+6d=15$........(3)
Now solving equation (2),

$a+(25-1)d=3[a+(8-1\left)d\right]$

$\therefore 2a-3d=0$.........(4)
Multiplying equation (4) by 2, we get $4a-6d=0$......(5)

On solving equations (3) and (5) simultaneously we get $a=3$

Substituting $a=3$ in equation (4)

$2\left(3\right)-3d=0$

$\therefore d=2$