Let f: N N be a function satisfying f(x+y) = 1 2 f(x) f(y) for all x, y N . If f(1) = 4 and _ k=1 ¹⁰ k f(k) = 2¹² + 4 , then the value of is

Step-by-step solution

Given the functional equation f(x+y) = 1 2 f(x) f(y) and f(1) = 4 . Substituting x=1, y=1 : f(2) = 1 2 f(1) f(1) = 1 2 (4)(4) = 8 = 2^3 Substituting x=2, y=1 : f(3) = 1 2 f(2) f(1) = 1 2 (8)(4) = 16 = 2^4 By observing the pattern, we can deduce f(k) = 2^ k+1 . We are given the sum S = _ k=1 ¹⁰ k f(k) = _ k=1 ¹⁰ k 2^ k+1 . This is an Arithmetico-Geometric Progression (AGP): S = 1 2^2 + 2 2^3 + 3 2^4 + + 10 2¹¹ Multiply by the common ratio 2 : 2S = 1 2^3 + 2 2^4 + + 9 2¹¹ + 10 2¹² Subtracting the second equation from