Respuesta :

3/2 = 3 x 1/2 
5/2 = 5 x 1/2 
so f(x) = 3 x 1/2 x (5 x 1/2) ^ x-1
f(x) = 3 x 1 x (5 x 1/2)^x-1   [notice the 1/2 is together]
          -----------------------
               2

but x/2 = x/1 divided by 2/1 to divide these, invert and multiply: x/2 x 2/1

so f(x) = 3 x 1 x (5/2 )^x-1
              ----------------------
                         2
     f(x) = 3 (5/2 ) ^ x-1 (x 1/2)

   f(x) = 1.5 (5/2)^x-1

 

Answer:

The equivalent function is  [tex]f(x)=\dfrac{3}{5}\cdot (\dfrac{5}{2})^{x}[/tex]

Step-by-step explanation:

Given: Pablo generates the function  [tex]f(x)=\dfrac{3}{2}\cdot (\dfrac{5}{2})^{x-1}[/tex] to determine the xth number in a sequence.

To find : The equivalent representation?

Solution :

[tex]f(x)=\dfrac{3}{2}\cdot (\dfrac{5}{2})^{x-1}[/tex]

We will re-write the function,

We distribute the exponent x-1    

[tex]f(x)=\dfrac{3}{2}\cdot (\dfrac{5}{2})^{x}\cdot (\dfrac{5}{2})^{-1}[/tex]

[tex]\because a^{m+n}=a^m\cdot a^n[/tex]

[tex]f(x)=\dfrac{3}{2}\cdot (\dfrac{5}{2})^{x}\cdot \dfrac{2}{5}[/tex]

[tex]\because (\dfrac{a}{b})^{-1}=\dfrac{b}{a}[/tex]

[tex]f(x)=\dfrac{3}{2}\cdot \dfrac{2}{5}\cdot (\dfrac{5}{2})^{x}[/tex]

[tex]f(x)=\dfrac{3}{5}\cdot (\dfrac{5}{2})^{x}[/tex]

Hence, The equivalent function is

 [tex]f(x)=\dfrac{3}{5}\cdot (\dfrac{5}{2})^{x}[/tex]

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