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Question
If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.
Solution
\[\text { We have: } \]
\[ S_n = nP + \frac{1}{2}n(n - 1)Q\]
\[\text { For } n = 1, S_1 = P + 0 = P\]
\[\text { For } n = 2, S_2 = 2P + Q \]
\[\text { Also }, a_1 = S_1 = P, \]
\[ a_2 = S_2 - S_1 \]
\[ = 2P + Q - P = P + Q\]
\[ \therefore d = a_2 - a_1 = P + Q - P = Q \]
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