A function f(x) is given by f(x) = 5ˣ/(5ˣ + 5), then the sum of the series f( 1/20 ) + f( 2/20 ) + f( 3/20 ) + ....... + f( 39/20 ) is equal to :

Solution

Maths sir · 4 steps

The inputs run from 1/20 to 39/20, so pair x with 2 − x and each pair adds to exactly 1.

1. Spot the pairing. The first and last inputs are 1/20 and 39/20, which add to 2. The same is true for 2/20 and 38/20, and so on. So find f(2-x).
2. Simplify f(2 − x). f(2-x) = 5²⁻ˣ/(5²⁻ˣ+5). Multiply the top and bottom by 5ˣ to get 25/(25 + 5 × 5ˣ) = 5/(5 + 5ˣ).

   f(x) + f(2-x) = 5ˣ/(5ˣ+5) + 5/(5ˣ+5) = 1

3. Count the pairs. The terms k = 1 to 19 pair with k = 39 down to 21. That makes 19 pairs, each worth 1. The middle term k = 20 has no partner, and f(1) = 5/(5+5) = 1/2.
4. So. The sum is 19 + 1/2 = 39/2, option (a).

The idea to keep. When the inputs are spread evenly and the first and last add to a constant c, check f(x) + f(c-x). It is usually a constant, and then the whole sum is that constant times the number of pairs.

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