Question:
Suppose that a function $f: \mathrm{R} \rightarrow \mathrm{R}$ satisfies $f(x+y)=f(x) f(y)$
for all $x, y \in \mathrm{R}$ and $f(1)=3$. If $\sum_{i=1}^{\mathrm{n}} f(i)=363$, then $\mathrm{n}$ is equal to_________.
Solution:
$\because f(x+y)=f(x) \cdot f(y) \quad \forall x \in \mathrm{R}$ and $f(1)=3$
$\Rightarrow f(x)=3^{x} \Rightarrow f(i)=3^{i}$
$\Rightarrow \sum_{i=1}^{n} f(i)=363 \Rightarrow 3+3^{2}+3^{3}+\ldots .+3^{n}=363$
$\Rightarrow \frac{3\left(3^{n}-1\right)}{3-1}=363 \quad\left[\because S_{n}=\frac{a\left(r^{n}-1\right)}{(r-1)}\right]$
$\Rightarrow 3^{n}-1=\frac{363 \times 2}{3}=242$
$\Rightarrow 3^{n}=243=3^{5} \Rightarrow n=5$