If f is a function satisfying `f(x+y)=f(x)f(y)`for all `x ,y in X`such that `f(1)=3`and `sum_(x=1)^nf(x)=120`, find the value of n.

Options

A. 4
B. 5
C. 6
D. none of these

Answer

A

Explanation

We have,
`f(x+y)=f(x)f(y)" for all "x,yinN`.
`:." "f(x)={f(1)}^(x)=3^(x)" "[becausef(1)=3]`
Now,
`underset(x=1)overset(n)sumf(x)=120`
`rArr" "underset(x=1)overset(n)sum3^(x)=120`
`rArr" "(3(3^(n)-1))/((3-1))=120rArr3^(n)-1=80rArr3^(n)=3^(4)rArrn=4`