If [x] be the greatest integer less than or equal to x,
then Σₙ₌₈¹⁰⁰ [ ( - 1)ⁿn/2 ] is equal to :

Solution

Maths sir · 4 steps

Split even and odd n, then pair each even term with the odd term after it. Every pair gives -1.

1. Even n. For even n, (-1)ⁿ = 1, so the term is [n/2] = n/2, already an integer. With n = 2k, the term is k.
2. Odd n. For odd n = 2k+1, the term is [-(2k+1)/2] = [-k - 0.5] = -(k+1). The greatest integer function moves a negative non-integer down, not towards zero.
3. Pair them up. Pair n = 2k with n = 2k+1: each pair gives k - (k+1) = -1. The pairs run from (8, 9) to (98, 99), so k = 4 to 49, which is 46 pairs. Only n = 100 is left over, and it gives 50.

   Σₙ₌₈¹⁰⁰[((-1)ⁿ n)/2] = 46 × (-1) + 50 = 4

4. So. The sum is 4, which is option (b).

The idea to keep. For a negative non-integer, [x] goes one step further down: [-4.5] = -5, not -4. With alternating signs, pair neighbouring terms and look for a constant pair sum.

Answer from the NTA final answer key. Solution by PrepWiser.