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If-the-sum-of-n-terms-f-a-series-is-A-n-2-B-n-where-A-B-are-constants-then-it-is-an-AP-

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Question Number 25741 by gugalesweta@gmail.com last updated on 13/Dec/17
If the sum of n terms f a series is A n^2 +B n , where A, B are constants, then it is an AP.
IfthesumofntermsfaseriesisAn2+Bn,whereA,Bareconstants,thenitisanAP.
Commented by math solver last updated on 13/Dec/17
yes.
yes.
Answered by Rasheed.Sindhi last updated on 14/Dec/17
S_n =An^2 +Bn S_(n−1) =A(n−1)^2 +B(n−1) T_n =(An^2 +Bn)−{A(n−1)^2 +B(n−1)} =An^2 −A(n−1)^2 +Bn−B(n−1) =An^2 −A(n^2 −2n+1)+Bn−Bn+B =2An−A+B The difference between consecutive terms d=T_k −T_(k−1) =2Ak−A+B−{2A(k−1)−A+B} =2Ak−A+B−2A(k−1)+A−B =2Ak−2Ak+2A =2A Since the difference between any two consecutive terms is constant The given sequence is AP.
Sn=An2+BnSn−1=A(n−1)2+B(n−1)Tn=(An2+Bn)−{A(n−1)2+B(n−1)}=An2−A(n−1)2+Bn−B(n−1)=An2−A(n2−2n+1)+Bn−Bn+B=2An−A+BThedifferencebetweenconsecutivetermsd=Tk−Tk−1=2Ak−A+B−{2A(k−1)−A+B}=2Ak−A+B−2A(k−1)+A−B=2Ak−2Ak+2A=2ASincethedifferencebetweenanytwoconsecutivetermsisconstantThegivensequenceisAP.
Commented by mrW1 last updated on 14/Dec/17
That′s great!
That′sgreat!

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