Let f(x) = _ k=1 ²⁰ k x^k . If the value of f'(1/2) is expressed in the form A - B 2^ -C , where A, B , and C are positive integers and B is an odd integer, then the value of A + B + C is equal to ______.

Step-by-step solution

Given f(x) = _ k=1 ²⁰ k x^k Let S = f'(1/2) = _ k=1 ²⁰ k^2 (1/2)^ k-1 We have S = 1 + 4(1/2) + 9(1/2)^2 + + 400(1/2)¹⁹ 1 2 S = 1(1/2) + 4(1/2)^2 + + 400(1/2)²⁰ Subtracting the two equations: 1 2 S = 1 + 3(1/2) + 5(1/2)^2 + + 39(1/2)¹⁹ - 400(1/2)²⁰ Let T = 1 + 3(1/2) + 5(1/2)^2 + + 39(1/2)¹⁹ 1 2 T = 1(1/2) + 3(1/2)^2 + + 37(1/2)¹⁹ + 39(1/2)²⁰ Subtracting again: 1 2 T = 1 + 2(1/2) + 2(1/2)^2 + + 2(1/2)¹⁹ - 39(1/2)²⁰ 1 2 T = 1 + 2 ( 1/2 - (1/2)²⁰ 1 - 1/2 ) - 39(1/2)²⁰ 1 2 T = 1 + 2(1 - 2⁻¹⁹) - 39 2⁻²⁰ = 3 - 4 2⁻²⁰ - 3