Suppose that a function f: R → R satisfies f (x + y) = f (x)f(y)...

$f\left(x\right) = {a}^{x}$

$⇒ f\left(1\right) = a = 3$

so $f\left(x\right) = {3}^{x}$

$\mathop{∑}_{i = 1}^{n}f\left(i\right) = 363$

$⇒ 3 + {3}^{2} + . . . . . {3}^{n} = 363$

$\dfrac{3\left({3}^{n} - 1\right)}{2} = 363$

${3}^{n} = 243 ⇒ n = 5$