Suppose that a function f:RR satisfies f(x+y)=f(x)f(y) for all x,yR, and f(1)=3.

Question

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

Answer: 5

Accepted Answer

$f(x)=3^x$, so $\sum_{i=1}^n 3^i = \dfrac{3(3^n-1)}{2}=363 \Rightarrow 3^n-1=242 \Rightarrow 3^n=243=3^5 \Rightarrow n=5.$

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