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  • If f(x) is a function satisfying f(x+y)=f(x).f(y), for all x, y belong to N , such that f(1)=3 and sigma f(x)=120 , then n equals

If f(x) is a function satisfying f(x+y)=f(x).f(y), for all x, y belong to N , such that f(1)=3 and sigma f(x)=120 , then n equals

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Solution:   We have ,     f(x+y)=f(x).f(y) 

                  and    f(1)=3    ,sum_x=1^nf(x)=120.

                Rightarrow    f(x+y)=f(x)cdot f(y)      Rightarrow  f(x)=a^kx

              Rightarrow       f(1)=3  gives    a^k=3Rightarrow f(x)=3^x

           Now ,         sum_x=1^nf(x)=120         Rightarrow        frac3(3^n-1)3-1=120

              Rightarrow                  3^n=81Rightarrow n=4.         

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Deependra Verma

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