JEE Main 2020 (Online) 6th September Morning ShiftQuestion 1
If $f(x+y)=f(x)f(y)$ and $\displaystyle\sum_{x=1}^{\infty}f(x)=2$, $x,y\in\mathbf{N}$, where $\mathbf{N}$ is the set of all natural numbers, then the value of $\dfrac{f(4)}{f(2)}$ is :
Sequence and Series · Geometric Progression
Solution
Formulas used. (1) $f(x+y)=f(x)f(y)$ on $\mathbf N$ gives $f(x)=k^x$, where $k=f(1)$(2) $\sum_{x\ge1}k^x=\frac{k}{1-k}$ for $|k|<1$Step 1 - Form of $f$. $f(x)=k^x$. Step 2 - Find $k$. $$\frac{k}{1-k}=2\ \Longrightarrow\ k=2-2k\ \Longrightarrow\ k=\frac23 .$$ Step 3. $$\frac{f(4)}{f(2)}=\frac{k^4}{k^2}=k^2=\frac49 .$$ Answer: (D)