Question

Pablo generates the function f(x) = 3/2(5/2)^x-1 to determine the xth number in a sequence. Which is an equivalent representation?

Answer 1

The given function is :

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

Two functions are said to be equivalent if it is written in another way but for every value of variable the value of function remains same.

$f(x)=3/2(5/2)^x-1\\\\ f(x)=3\times 5^x\times \frac{1}{2^{x+1}} -1[\text{used law of exponent,} x^a\times x^b=x^{a+b}]$

Answer 2

Answer:

The equivalent function is $f(x)=\dfrac{3}{5}\cdot (\dfrac{5}{2})^{x}$

Step-by-step explanation:

Given: $f(x)=\dfrac{3}{2}\cdot (\dfrac{5}{2})^{x-1}$

This function represents xth number of the sequence.

We need to re-write the function which is equivalent to given one.

First we distribute the exponent x-1

$f(x)=\dfrac{3}{2}\cdot (\dfrac{5}{2})^{x}\cdot (\dfrac{5}{2})^{-1}$                                     $\because a^{m+n}=a^m\cdot a^n$

$f(x)=\dfrac{3}{2}\cdot (\dfrac{5}{2})^{x}\cdot \dfrac{2}{5}$                                               $\because (\dfrac{a}{b})^{-1}=\dfrac{b}{a}$

$f(x)=\dfrac{3}{2}\cdot \dfrac{2}{5}\cdot (\dfrac{5}{2})^{x}$

$f(x)=\dfrac{3}{5}\cdot (\dfrac{5}{2})^{x}$

Hence, The equivalent function is $f(x)=\dfrac{3}{5}\cdot (\dfrac{5}{2})^{x}$