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Question
If the sum of first p terms of an AP is 2 (ap2 + bp), find its common difference.
Solution
Let Sp denotes the sum of first p terms of the AP.
∴ sp = 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,
pth 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.
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