"If f(x + y) = f(x)f(y) and $\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$ , x, y $\in$ N, where N is the set of all natural number, then the value of ${{f\left( 4 \right)} \over {f\left( 2 \right)}}$ is :"

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Accepted Answer

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

$\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$

$\Rightarrow$ f(1) + f(2) + f(3) + ........$\infty$ = 2 ....(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). 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 + ....... = 2

$\Rightarrow$ ${{f\left( 1 \right)} \over {1 - f\left( 1 \right)}}$ = 2

$\Rightarrow$ f(1) = ${2 \over 3}$

Now f(2) = ${\left( {{2 \over 3}} \right)^2}$

and f(4) = ${\left( {{2 \over 3}} \right)^4}$

$\therefore$ ${{f\left( 4 \right)} \over {f\left( 2 \right)}}$

= $${{{{\left( {{2 \over 3}} \right)}^4}} \over {{{\left( {{2 \over 3}} \right)}^2}}}$$ = ${4 \over 9}$"

If f(x + y) = f(x)f(y) and $\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$ , x, y $\in$ N, where N is the set of all natural number, then the value of ${{f\left( 4 \right)} \over {f\left( 2 \right)}}$ is :

Solution

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If f(x + y) = f(x)f(y) and $\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$ , x, y $\in$ N, where N is the set of all natural number, then the value of ${{f\left( 4 \right)} \over {f\left( 2 \right)}}$ is :

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