Let f(x)=(2ˣ⁺²+16)/(2²ˣ⁺¹+2ˣ⁺⁴+32). Then the value of 8(f(1/15)+f(2/15)+…+f(59/15)) is equal to

Solution

Maths sir · 4 steps

Simplify f to 2/(2ˣ+4), then pair f(x) with f(4-x): each pair adds to 1/2.

1. Simplify f. Put t = 2ˣ. The numerator is 4t + 16 = 4(t+4). The denominator is 2t² + 16t + 32 = 2(t+4)². So f(x) = 2/(2ˣ + 4).
2. Find the pairing. The arguments run from 1/15 to 59/15, and k/15 + (60-k)/15 = 4. So test f(4-x). Since 2⁴⁻ˣ = 16/t, you get f(4-x) = 2/(16/t + 4) = t/2(t+4).

   f(x) + f(4-x) = 2/(t+4) + t/2(t+4) = (4+t)/2(t+4) = 1/2

3. Count the pairs. Pair k with 60-k for k = 1 to 29. That gives 29 pairs, each worth 1/2. The middle term k = 30 is f(2) = 2/(4+4) = 1/4, and it has no partner.
4. So. The sum is 29/2 + 1/4 = 59/4, and 8 × 59/4 = 118, option (c).

The idea to keep. For a sum over evenly spaced arguments, add the first and last argument to get a. If f(x) + f(a-x) is constant, pair terms from both ends and do not forget the unpaired middle term.

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