Test Index

Functions Part 1

Section: Mathematics

© examsnet.com

Question : 54 of 100

Marks: +1, -0

If

f(x+y)=f(x)\cdot f(y)

\sum\limits_{x=1}^{\infty} f(x)=2, x, y \in N

N

is the set of all natural numbers, thenthe value of

\frac{f(4)}{f(2)}

[6 Sep 2020 Shift 1]

$\frac{1}{9}$
$\frac{4}{9}$
$\frac{1}{3}$
$\frac{2}{3}$

Solution:

f(x+y)=f(x) f(y)

\;\sum\limits_{x=1}^{\infty} f(x)=2

\Rightarrow f(1)+f(2)+f(3)+\ldots =2 \ldots (1)

On

f(x+y)=f(x) f(y)

* Put

x=1, y=1

f(2)=(f(1))^{2}

* Put

x=2, y=1

f(3)=f(2) \cdot f(1)=f((1))^{3}

* Put

x=2, y=2

f(4)=f((2))^{2}=f((1))^{4}

Now put these values in equation (1)

f(1)+f((1))^{2}+f((1))^{3}+\ldots =2

\Rightarrow \;\frac{f(1)}{1-f(1)}=2

\Rightarrow f(1)=\;\frac{2}{3}

Now

f(2)=\left(\;\frac{2}{3}\right)^{2}

and

f(4)=\left(\;\frac{2}{3}\right)^{4}

\therefore \;\frac{f(4)}{f(2)}

=\;\frac{\left(\frac{2}{3}\right)^{4}}{\left(\;\frac{2}{3}\right)^{2}}=\;\frac{4}{9}

© examsnet.com

Go to Question:

Prev Question

Next Question