If f(x)=2ˣ/(2ˣ+√2), x ∈ ℝ, then Σk=1⁸¹ f(k/82) is equal to

Solution

Maths sir · 4 steps

The terms k and 82 − k add to 1, so pair the sum from both ends and the 81 terms collapse.

1. Find the pairing. The inputs k/82 and (82-k)/82 add to 1. So work out f(1-x) and add it to f(x).

   f(1-x) = 2¹⁻ˣ/(2¹⁻ˣ+√2) = 2/(2+√2 2ˣ) = √2/(2ˣ+√2) ⇒ f(x)+f(1-x)=1

2. Count the pairs. Pair k=1 with k=81, k=2 with k=80, and so on up to k=40 with k=42. That gives 40 pairs, each adding to 1, so together they give 40.
3. Middle term. k=41 has no partner because 41/82=1/2. Here f(1/2)=√2/(√2+√2)=1/2.
4. So. The sum is 40+1/2=81/2, which is option (d).

The idea to keep. When a sum runs over k/n, check f(x)+f(1-x) first. If it is a constant, pair the terms from both ends, and do not forget the unpaired middle term when the count is odd.

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