Question

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

Answer 1

Let the First term of $A.P=a$

and the common diff $=d$

Now, as given

The sum of $5^{th}$ and $9^{th}$ term $=30$

$T_{5^{th}}+T_{9^{th}}=30$

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

$2a+12d=30$

$a+6d=15......\left(1\right)$

Ans Also given

$T_{{25}^{th}}=3T_{8^{th}}$

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

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

[Eqn $\left(1\right)\times 2$]-Eqn $\left(2\right)$

$2a+12d=30$

$2a-3d=0$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$15d=30$

$d=2$

From Eqn $\left(2\right)$

$1a-3\left(2\right)=0$

$a=3$

Hence the series $\left(AP\right)$ is

$=a,a+d,a+2d$

$=3,5,\mathrm{7..........}$