Question

If f (x) is a function satisfying $f(x+y)=f\left(x\right).f\left(y\right)$ for all $x,yϵ$ N such that f(1) = 3 and ${\sum }_{x=1}^{n}f\left(x\right)=120$. Then,the value of n is

• A. 4
• B. 5
• C. 6
• D. None of these

Answer 1

The correct option is A 4

Given, f(x+y) = f(x) f(y) for all $x,yϵN$
$\therefore ForanyxϵN,f\left(x\right)=\left[f\right(1){]}^x=3^x[\therefore f\left(1\right)=3]$
Since, ${\sum }_{x=1}^{n}f\left(x\right)=120$
$\Rightarrow 3^1+3^2+3^3+.....+3^n=120$
$\Rightarrow 3^n-1=80$
$\Rightarrow 3^n=81=3^4\Rightarrow n=4$