Let R denote the set of all real numbers. Let f: R R be a function such that f(x)>0 for all x R , and f(x+y)=f(x) f(y) for all x, y R . Let the real numbers a ₁, a ₂ , a ₅₀ be in an arithmetic progression. If f ( a ₃₁ )=64 f ( a ₂₅ ) , and _ i=1 ⁵⁰ f (a_i )=3 (2²⁵+1 ) then the value of _ i=6 ³⁰ f (a_i ) is ________

Step-by-step solution

aligned & f(x+y)=f(x) f(y) & f(x)=k^x (f(x)>0 x R) & f (a₃₁ )=64 f (a₂₅ ) aligned aligned & k ^ ( a +30 ~d ) =64 k ^ ( a +24 ~d ) & k ^ 6 ~d =64 & k ^ d =2 aligned _ i =1 ⁵⁰ f ( a _ i )= f ( a ₁ )+ f ( a ₂ )+ + f ( a ₅₀ ) aligned & = k ^ a + k ^ a + d + + k ^ a +49 ~d = k ^ a ( k ^ 50 ~d -1 ) k ^ d -1 & = k ^ a (2⁵⁰-1 )=3 (2²⁵+1 )( Given ) & k ^ a = 3 2²⁵-1 aligned aligned & _ i =6 ³⁰ f ( a _ i )= k ^ a +5 ~d + k ^ a +6 ~d + + k ^ a +29 ~d & = k ^ a +5 ~d ( k ^ 25 ~d -1 ) k ^ d -1 = k ^ a ( k ^ d )^5 (2²⁵-1 ) & = 3