Mathematics - Functions Question with Solution | TestHub

Given,

$f\left(x+y\right)=f\left(x\right)·f\left(y\right)$ and $f\left(1\right)=3$ 

Now taking $x=1 \& y=1$ we get,

$f\left(1+1\right)=f\left(1\right)·f\left(1\right)\Rightarrow f\left(2\right)=3^2$

Similarly $f\left(2+1\right)=f\left(2\right)·f\left(1\right)\Rightarrow f\left(3\right)=3^2\times 3=3^3$

And so on $f\left(n\right)=3^n$

So, $\sum _{r=1}^{n}f\left(r\right)=3279$

$\Rightarrow f\left(1\right)+f\left(2\right)+f\left(3\right).......f\left(n\right)=3279$

$\Rightarrow 3+3^2+3^3.......+3^n=3279$

$\Rightarrow 3\times \frac{3^n-1}{3-1}=3279$

$\Rightarrow \frac{3^n-1}{2}=1093$

$\Rightarrow 3^n-1=2186$

$\Rightarrow 3^n=2187$

$\Rightarrow 3^n=3^7$

$\Rightarrow n=7$