Question

If the sum of first p term of an A.P. is ap² + bp, find its common difference.

 

Answer 1

Here, we are given,

 S_p = ap² + 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 .