If Sn = nP + n/2 (n – 1)Q, where Sn denotes the sum of first n terms of an A.P., then the common difference of the A.P. is

Question

If S_n = nP + \(\frac{n}{2}\)(n – 1)Q, where S_n denotes the sum of first n terms of an A.P., then the common difference of the A.P. is

(a) Q 

(b) P + Q 

(c) P + 2Q 

(d) 2P + 3Q

Answer 1

Answer : (a) Q

nth term of an A.P. is given by 

T_n = Sum of n terms – Sum of (n – 1) terms

= S_n – S_{n – 1}

= nP + \(\frac{n}{2}\) (n – 1)Q – \(\big[(n-1)P +\frac{(n-1)}{2}(n-2)Q\big]\) 

= nP – nP + P + \(\frac{Q}{2}\)[n² – n – n² + 3n – 2] 

= P + \(\frac{Q}{2}\) (2n – 2) 

= P + (n – 1) Q. 

∴ Common difference (d) = T_n – T_{n – 1} 

= P + (n – 1)Q – (P + (n – 2)Q) 

= Q.