Paragraph: Let f(x) be a function satisfying f(x+y)=f(x) f(y) for all x, y N such that f(1)=2 : Question: If _ x=2 ^n f(x)=2044 , then what is the value of n ?

Step-by-step solution

Given that f(x+y)=f(x) f(y) and f(1)=2 aligned & f(2)=f(1+1)=f(1) f(1)=2.2=4 & f(3)=f(1+2)=f(1) f(2)=2.4=8 & f(4)=f(1+3)=f(1) f(3)=2.8=16, So on aligned Now, _ x=2 ^n f(x)=2044 aligned & f(2)+f(3)+f(4)+ +f(n)=2044 & 4+8+16 (n-1) terms =2044 & 4 (2^ n-1 -1 ) 2-1 =2044 2^ n-1 -1=511 & 2^ n -1 =512=2^9 & n=10 aligned