If the sum of first p terms of an A.P., is $ap^{2} +bp$, find its common difference.

Given: The sum of first p terms$=ap^{2} +bp$

What to do: To find out its common difference.

Solution: 

Here the given A.P. has the sum of first p terms =$ap^{2} +bp$

Let us say first term of given A.P. is x and the common difference is y.

then the sum of the A.P. $=\frac{p}{2} \ [ 2x+( p-1) y]$

$\Rightarrow $ Given sum $ap^{2} +bp=\ \frac{p}{2}[ 2x+( p-1) y]$

$\Rightarrow  2ap^{2} +2bp=p[ 2x+( p-1) y]$

$\Rightarrow  2p( ap+b) =p[ 2x+( p-1) y]$

$\Rightarrow 2ap+2b=2x+( p-1) \ y$

$\Rightarrow \ \ \ \ \ 2b+2ap=( 2x-y) +\ py$

On comparing terms of both sides

$\Rightarrow 2a=y$

$2x-y=2b$

$\Rightarrow  2x=2b+y$

$\Rightarrow  2x=2b+2a$                                $( by\ putting\ the\ value\ 2a=y)$

$\Rightarrow x=a+b$

Thus common difference y=2

And first term $x=a+b$