Question

The sum of $5^{\text{th}}$ and $9^{\text{th}}$ terms of an AP is $72$ and sum of $7^{\text{th}}$ and ${12}^{\text{th}}$ terms is $97$. Find the AP.

Answer 1

Step 1. Given values:

$a_5+a_9=72$ and $a_7+a_{12}=97$

Step 2. Finding the AP:
We know that $a_n=a+(n-1)d$.

$\begin{array}{ccc}\left(a+4d\right)+(a+8d)=72& \text{and}& (a+6d)+(a+11d)=97\\ 2a+12d=72& \text{and}& 2a+17d=97\end{array}$

On solving the equations, we get,
$\begin{array}{cc}\Rightarrow & -5d=-25\\ \Rightarrow & d=5\end{array}$
On substituting the value of d in equation, we get,
$\begin{array}{cc}\Rightarrow & 2a=12\\ \Rightarrow & a=6\end{array}$
The general form of AP is $a,a+d,a+2d,a+3d,a+4d,\dots$
$\Rightarrow 6,11,16,21,26,\dots$

Therefore, the AP is $6,11,16,21,26,\dots$