Q12 of 176 Page 8

If the sum of n terms of an A.P. is nP + 1/2 n( n —1)Q , where P and Q are constants, find the common difference of the A.P.


Taking n = 1, we get



⇒ S1 = P


⇒ a1 = P


Taking n = 2, we get



⇒ S2 = 2P + Q


∴ a2 = S2 – S1 = 2P + Q – P = P + Q


Taking n = 3, we get



⇒ S3 = 3P + 3Q


∴ a3 = S3 – S2 = 3P + 3Q – 2P – Q = P + 2Q


So, a = P,


d = a2 – a1 = P + Q – (P) = Q


= a3 – a2 = P + 2Q – (P + Q) = P + 2Q – P – Q = Q


Hence, the common difference is Q.


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