# 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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