If the sum of the first m terms of an AP is equal to the sum of its first n terms (m ≠ n), what is the sum of its first (m + n) terms?

Answer and explanation

Correct answer: B. 0

  1. Step 1: S_m = (m/2)[2a + (m - 1)d] and S_n = (n/2)[2a + (n - 1)d].
  2. Step 2: S_m = S_n => 2a(m - n) + d[m(m - 1) - n(n - 1)] = 0.
  3. Step 3: 2a(m - n) + d(m^2 - n^2 - (m - n)) = 0 => (m - n)[2a + (m + n - 1)d] = 0.
  4. Step 4: Since m ≠ n, 2a + (m + n - 1)d = 0.
  5. Step 5: S_(m + n) = ((m + n)/2)[2a + (m + n - 1)d] = ((m + n)/2) * 0 = 0.
Why not the others: m + n, -(m + n) and 1 do not fit.

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