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

#### Solution

Here, we are given,

S_{p} = ap^{2} + bp

Let us take the first term as *a’* and the common difference as *d*.

Now, as we know,

a_{p} = S_{p} - S_{p -1}

So, we get,

`a_p = (ap^2 +bp) - [a(p - 1)^2 + b (p-1)]`

`= ap^2 + bp - [a(p^2 + 1 -2 p) + bp - b] [\text{Using} (a - b)^2 = a^2 + b^2 - ab]`

` = ap^2 + bp - (ap^2 + a - 2ap + bp -b)`

`=ap^2 + bp - ap^2 - a + 2ap - bp +b`

`=2ap - a + b` ..............(1)

Also,

`a_p = a' + (p-1)d`

`= a' + pd - d `

`= pd + ( a' - d)` ..............(2)

On comparing the terms containing *p *in (1) and (2), we get,

dp = 2ap

d = 2a

Therefore, the common difference is d = 2a .