Question

Suppose that a function $f: R \to R$ satisfies $f(x+y)=f(x) f(y)$ for all $x, y \in R$ and $f(1)=3$. If $\displaystyle\sum_{i=1}^{n} f(i)=363,$ then $n$ is equal to _____.

Answer 1

$f(x+y)=f(x) f(y)$
put $x=y=1 \quad f(2)=(f(1))^{2}=3^{2}$
put $x=2, y=1 \quad f(3)=(f(1))^{3}=3^{3}$
$\vdots$
Similarly $f(x)=3^{x}$
$\displaystyle\sum_{i=1}^{n} f(i)=363$
$ \Rightarrow \displaystyle\sum_{i=1}^{n} 3^{i}=363$
$\left(3+3^{2}+\ldots+3^{n}\right)=363$
$\frac{3\left(3^{n}-1\right)}{2}=363$
$3^{n}-1=242 \Rightarrow 3^{n}=243$
$\Rightarrow n=5$