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.

Answer 1

$$\begin{gathered} \mathrm{W}e\mathrm{have}: \\ {S}_n=nP+\frac{1}{2}n(n-1)Q \\ \mathrm{For}n=1,S_1=P+0=P \\ \mathrm{For}n=2,S_2=2P+Q \\ \mathrm{Also},a_1=S_1=P, \\ {a}_2=S_2-S_1 \\ =2P+Q-P=P+Q \\ \therefore d=a_2-a_1=P+Q-P=Q \end{gathered}$$