Question

La función f satisface f(x + y) = f(x) • f(y) y f(1) = 2

$\frac{f(2)}{f(1)} + \frac{f(3)}{f(2)} ........ + \frac{f(2021)}{f(2020)}$
cuanto vale ?​

Answer 1

Aplicaremos la condición inicial para calcular lo que nos piden

      $\begin{array}{c}\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots+\dfrac{f(2021)}{f(2020)}=\dfrac{f(1+1)}{f(1)}+\dfrac{f(1+2)}{f(2)}+\cdots\dfrac{f(1+2020)}{f(2020)}}\\\\\sf{Aplicamos\ f(x+y) = f(x)\cdot f(y)}\\\\\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots+\dfrac{f(2021)}{f(2020)}=\dfrac{f(1)\cdot f(1)}{f(1)}+\dfrac{f(1)\cdot f(2)}{f(2)}+\cdots+\dfrac{f(1)\cdot f(2020)}{f(2020)}}\\\\\sf{Simplificamos}\end{array}$

         $\begin{array}{c}\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots+\dfrac{f(2021)}{f(2020)}=\dfrac{f(1)\cdot \not \!\!f(1)}{\not \!\!f(1)}+\dfrac{f(1)\cdot \not \!\!f(2)}{\not \!\!f(2)}+\cdots+\dfrac{f(1)\cdot \not \!\!f(2020)}{\not \!\!f(2020)}}\\\\\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots\dfrac{f(2021)}{f(2020)}=\underbrace{\sf{f(1)+f(1)+\cdots +f(1)}}_{\sf{2020\ veces}}}\\\\\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots\dfrac{f(2021)}{f(2020)}=2020\cdot f(1)}\\\\\end{array}$

                               $\begin{array}{c}\sf{Pero\ f(1)=2, entonces}\\\\\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots\dfrac{f(2021)}{f(2020)}=2020\cdot f(1)}\\\\\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots\dfrac{f(2021)}{f(2020)}=2020(2)}\\\\\boxed{\boxed{\boldsymbol{\red{\sf{\dfrac{f(2)}{f(1)}+\dfrac{f(3)}{f(2)}+\cdots\dfrac{f(2021)}{f(2020)}=4040}}}}}\end{array}$

                                            $\boxed{\sf{{R}}\quad\raisebox{10pt}{ \sf{\red{O}} }\!\!\!\!\raisebox{-10pt}{ \sf{\red{O}} }\quad\raisebox{15pt}{ \sf{{G}} }\!\!\!\!\raisebox{-15pt}{ \sf{{G}} }\quad\raisebox{15pt}{ \sf{\red{H}} }\!\!\!\!\raisebox{-15pt}{ \sf{\red{H}} }\quad\raisebox{10pt}{ \sf{{E}} }\!\!\!\!\raisebox{-10pt}{ \sf{{E}} }\quad\sf{\red{R}}}\hspace{-64.5pt}\rule{10pt}{.2ex}\:\rule{3pt}{1ex}\rule{3pt}{1.5ex}\rule{3pt}{2ex}\rule{3pt}{1.5ex}\rule{3pt}{1ex}\:\rule{10pt}{.2ex}$

Answer 2

Respuesta:

sale 65132 porque 46+454.0 = 231325

Explicación paso a paso: