જો f( x + y )=f( x ) f( y ) અને sum limits_{ | vedclass

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Question

$f(x+y)=f(x) \cdot f(y)$

$\sum_{x=1}^{\infty} f(x)=2 \quad$ where $x, y \in N$

$f(1)+f(2)+f(3)+\ldots . \infty=2 \ldots .(1)($ Given $)$

Now

Vedclass · Accepted Answer

$\frac{4}{9}$

Vedclass · Answer

$f(x+y)=f(x) \cdot f(y)$

$\sum_{x=1}^{\infty} f(x)=2 \quad$ where $x, y \in N$

$f(1)+f(2)+f(3)+\ldots . \infty=2 \ldots .(1)($ Given $)$

Now for $f(2)$ put $x=y=1$

$f(2)=f(1+1)=f(1) \cdot f(1)=(f(1))^{2}$

$f(3)=f(2+1)=f(2) \cdot f(1)=(f(1))^{3}$

Now put these values in equation

$f(1)+(f(1))^{2}+\left[f(1)^{2}+\ldots \infty=2\right]$

$\frac{f(1)}{1-f(1)}=2$

$\Rightarrow f(1)=\frac{2}{3}$

Now $f(2)=\left(\frac{2}{3}\right)^{2}$

$f(4)=\left(\frac{2}{3}\right)^{4}$

then the value of $\frac{f(4)}{f(2)}=\frac{\left(\frac{2}{3}\right)^{4}}{\left(\frac{2}{3}\right)^{2}}=\frac{4}{9}$

જો $f( x + y )=f( x ) f( y )$ અને $\sum \limits_{ x =1}^{\infty} f( x )=2, x , y \in N$ જ્યાં $N$ એ બધી પ્રાકૃતિક સંખ્યાઓનો ગણ હોય તો $\frac{f(4)}{f(2)}$ ની કિમત શોધો

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