Let a1, a2, a3, …… , a100 be an arithmetic progression with a1 = 3 and SP = Σai, where the summation runs over p and 1 ≤ p ≤ 100. For any integer n with 1 ≤ n ≤ 20, let m = 5n. What is the value of a2 if Sm/ Sndoes not depend on n.

Given that-a1, a2, a3, …… , a100 -APa1=3 and SP = ΣaiNow Sm/ Sn=(m/2(2a1+(m-1)d)/(n/2(2a1+(n-1)d)Now m=5nSo,Sm/Sn=(5n/2(2a1+(5n-1)d)/(n/2(2a1+(n-1)d)=(5(2a1+(5n-1)d)/((2a1+(n-1)d)Now d=(a2-a1)=(a2-3)=common diff.But Sm/Sn does not depend on nSo 5nd=0 and nd=0=5n(a2-3)=0This gives a2=3.