If an APa=1,Sn:(S2n-Sn)= constant, ∀n∈N then the common difference d=?
4
12
2
3
Explanation for the correct option:
Step 1. Find the common difference d:
Given, a=1 and Sn:(S2n-Sn)= constant
S1(S2-S1)=S2(S4-S2)
⇒ S1(S4-S2)=S2(S2-S1)
⇒S1S4-S1S2=S22-S1S2
⇒ S22=S1S4
⇒ (a+a+d)2=a42(2a+3d)
⇒ (2a+d)2=a(4a+6d)
Step 2. By Solving it, we get
∴d=2a=2(1)=2
Hence, Option ‘C’ is Correct.
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d=Sn−k Sn−1+Sn−2, then k =