Question

Let $f$ be a function satisfying $f(x+y)=f\left(x\right)\cdot f\left(y\right)$ for all $x,yϵR$. If $f\left(1\right)=3$ then $\sum _{r=1}^{n}f\left(r\right)$ is equal to

• A. $\frac{3}{2}(3^n-1)$
• B. $\frac{3}{2}n(n+1)$
• C. $3^{n+1}-3$
• D. none of these

Answer 1

The correct option is A $\frac{3}{2}(3^n-1)$

Let $f\left(x\right)=a^x$
Therefore
$f(x+y)$
$=a^{x+y}$
$=a^x{a}^y$
$=f\left(x\right).f\left(y\right)$
Now it is given that $f\left(1\right)=3$
Therefore $a=3$
Hence ${\sum }_{r=1}^{r=n}f\left(r\right)$
$=3+3^2+3^3+3^4+..{.3}^n$
The following summation is a G.P with a common ratio as 3 and no.of terms as n.
The sum of the G.P will be
$=\frac{3(3^n-1)}{3-1}$
$=\frac{3(3^n-1)}{2}$