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1.Relation and Function
medium
Suppose that a function $f: R \rightarrow R$ satisfies $f(x+y)=f(x) f(y)$ for all $x, y \in R$ and $f(1)=3 .$ If $\sum \limits_{i=1}^{n} f(i)=363,$ then $n$ is equal to
A
$6$
B
$5$
C
$7$
D
$4$
(JEE MAIN-2020)
Solution
$f(x+y)=f(x) f(y)$
put $x = y =1 \quad f(2)=(f(1))^{2}=3^{2}$
put $x=2, y=1 \quad f(3)=(f(1))^{3}=3^{3}$ :
Similarly $f(x)=3^{x}$
$\sum_{i=1}^{n} f(i)=363 \Rightarrow \sum_{i=1}^{n} 3^{i}=363$
$\left(3+3^{2}+\ldots+3^{n}\right)=363$
$\frac{3\left(3^{n}-1\right)}{2}=363$
$3^{n}-1=242 \Rightarrow 3^{n}=243$
$\Rightarrow n =5$
Standard 12
Mathematics
