Discussion
The sum of the first n terms of an A.P. is given by \(S_n = 3n^2 + 5n\). Find the common difference.
Answer: B
The nth term of an A.P. can be found by \(a_n = S_n - S_{n-1}\). First term \(a_1 = S_1 = 3(1)^2 + 5(1) = 8\). Sum of first two terms \(S_2 = 3(2)^2 + 5(2) = 12 + 10 = 22\). The second term \(a_2 = S_2 - S_1 = 22 - 8 = 14\). The common difference \(d = a_2 - a_1 = 14 - 8 = 6\). Alternatively, for a sum in the form \(An^2+Bn\), the common difference is \(2A\). Here A=3, so \(d=2 \times 3=6\).
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