Suppose that a function


Suppose that a function $f: \mathrm{R} \rightarrow \mathrm{R}$ satisfies $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f(\mathrm{y})$ for all $\mathrm{x}, \mathrm{y} \in \mathrm{R}$ and

$f(1)=3$. If $\sum_{\mathrm{i}=1}^{\mathrm{n}} f(\mathrm{i})=363$, then $\mathrm{n}$ is equal to



$f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) f(\mathrm{y})$

put $x=y=1 \quad f(2)=(f(1))^{2}=3^{2}$

put $x=2, y=1 \quad f(3)=(f(1))^{3}=3^{3}$

Similarly $f(\mathrm{x})=3^{\mathrm{x}}$

$\sum_{\mathrm{i}=1}^{\mathrm{n}} f(\mathrm{i})=363 \Rightarrow \sum_{\mathrm{i}=1}^{\mathrm{n}} 3^{\mathrm{i}}=363$



$3^{n}-1=242 \Rightarrow 3^{n}=243$

$\Rightarrow \mathrm{n}=5$

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