Question

If ${f_0}\left( x \right) = \dfrac{x}{{\left( {x + 1} \right)}}\,and\,{f_{n + 1}} = {f_0} \cdot {f_n}\left( x \right)$ and =0, 1, 2,... then ${f_n}\left( x \right)$ is

Answer 1

Hint: Start by putting the value of n=0 in $\,{f_{n + 1}} = {f_0} \cdot {f_n}\left( x \right)$, then put n=1 repeat the step 2 to 3 times observe the pattern of the answers and generalise while putting n=n.

Complete step-by-step answer:

Given, \[{f_0}\left( x \right) = \dfrac{x}{{x + 1}}\]
\[{f_{n + 1}}\left( x \right) = {f_0}\left( x \right) \cdot {f_n}\left( x \right) \to (1)\]
On putting the value of n = 0, we get,
\[ \Rightarrow {f_1}\left( x \right) = {f_0}\left( x \right) \cdot {f_0}\left( x \right)\]
On putting the values of ${f_0}\left( x \right)\,$, we get,
\[ \Rightarrow {f_1}(x) = {\left( {\dfrac{x}{{x + 1}}} \right)^2} \to (2)\]
On putting the value of n = 1 in equation 1, we get,
\[ \Rightarrow {f_2}\left( x \right) = {f_0}\left( x \right) \cdot {f_1}\left( x \right)\]
On putting the values of${f_0}\left( x \right)\,and\,{f_1}\left( x \right)$, we get,
\[ \Rightarrow {f_2}(x) = {\left( {\dfrac{x}{{x + 1}}} \right)^3}\]
On putting the value of n = 2 in equation (1) , we get,
\[ \Rightarrow {f_3}\left( x \right) = {f_0}\left( x \right) \cdot {f_2}\left( x \right)\]
On putting the values of ${f_0}\left( x \right)\,and\,{f_2}\left( x \right)$, we get,
\[ \Rightarrow {f_3}(x) = {\left( {\dfrac{x}{{x + 1}}} \right)^4}\]
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So, from above equations, we can generalize the ${f_n}\left( x \right)$as
\[ \Rightarrow {f_n}(x) = {\left( {\dfrac{x}{{x + 1}}} \right)^{n + 1}}\].
Let us verify the equation by substituting the value of n as 1 i.e..,
\[ \Rightarrow {f_1}(x) = {\left( {\dfrac{x}{{x + 1}}} \right)^{1 + 1}} = {\left( {\dfrac{x}{{x + 1}}} \right)^2} \to (3)\].
As we can see the value of ${f_1}(x)$ is the same in both equations (2) and (3).
Hence, \[{f_n}(x) = {\left( {\dfrac{x}{{x + 1}}} \right)^{n + 1}}\]is our required${f_n}\left( x \right)$.

Note: In these questions, the aim should be to get the pattern while equating the values which are known to us, once that is done a general equation can be formed such that different values can be put to get the desired result.