Question

Let f be a function satisfying $f\left(x+y\right)=f\left(x\right)f\left(y\right)\forall x,y\in R$. If $f\left(1\right)=3$ then $\sum _{r=1}^{n}f\left(r\right)$ is less than or equal to $(n>10)$

• A. $\frac{3}{2}\left(3^n-1\right)$
• B. $\frac{3}{2}n\left(n+1\right)$
• C. $3^{n+1}-3$
• D. $\frac{3}{2}n\left(n-1\right)$

Answer 1

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

$f(x+y)=f\left(x\right)f\left(y\right)f\left(1\right)=3\therefore f\left(x\right)=a^{x}f\left(1\right)=3a^1=3\therefore a=3{\sum }_{r=1}^{n}f\left(r\right)=3+3^2+3^3+....+3^n=\frac{3(3^n-1)}{3-1}=\frac{3}{2}(3^n-1)=\frac{3}{2}(3^n-1)$