Question

If the sum of first p terms of an AP is 2 (ap²  +  bp), find its common difference.

Answer 1

Let S_p  denotes the sum of first p terms of the AP.

 ∴ s_p  = ap²  + bp

⇒ `s_(p-1) = a (p-1)^2  + b( p-1)`

= a( p² - 2p +1 ) +b (p-1) 

= ap² - ( 2a -b) p+ (a-b)  

Now,

p^th  term of AP ` a_p = s_p - s_(p-1)`

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

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

= 2ap - (a-b) 

Let d be the common difference of the AP.

∴ d = `a_p - a_( p-1)`

= [ 2 ap - (a-b) ] = [ 2a (p-1) - (a-b) ]

= 2ap - (a-b) - 2a (p-1 ) + (a-b) 

= 2a

Hence, the common difference of the AP is 2a.