Let a₁, a₂, a₃, ..., a₁₀₀ be an arithmetic progression with a₁ = 3 and S₁₀₀ is the sum of 100 terms. For any integer n with 1 ≤ n ≤ 20, let m = 5n. If Sₘ/Sₙ does not depend on n, then a₂ is?

We know that the sum of n terms of an A.P. is given by
Sₙ = n/2 [2a + (n - 1)d]
Sₘ/Sₙ = [5n/2 [2a + (5n - 1)d]] / [n/2 [2a + (n - 1)d]] = [5 [2a + (5n - 1)d]] / [2a + (n - 1)d]
Given that a₁ = 3, a = 3.
S₅ₙ/Sₙ = [5 (6 + (5n - 1)d)] / [6 + (n - 1)d] = [30 + 25nd - 5d] / [6 + nd - d] = 5[6 + 5nd - d] / [6 + nd - d]
Since this is independent of n, then 6 - d = 0 => d = 6
Hence, the second term = first term + d = 3 + 6 = 9
Hence, the answer is 9.