The equation that matches the table is [tex]f(x)=(\frac{2}{3} )^{x}[/tex]
Explanation:
From the table, we can see that this is a geometric progression because the common difference in the y-term is [tex]\frac{2}{3}[/tex]
Thus, [tex]r=\frac{2}{3}[/tex] and [tex]a=\frac{2}{3}[/tex]
To determine the equation, let us substitute the values of r and a in the general form of geometric progression.
The general form of geometric progression is given by
[tex]a_{n}=a r^{n-1}[/tex]
Now, substituting we have,
[tex]a_{n}=\frac{2}{3}\left(\frac{2}{3}\right)^{n-1}[/tex]
Simplifying by adding the powers of similar terms, we get,
[tex]a_{n}=(\frac{2}{3})^{n}[/tex]
Writing it in terms of x, we get,
[tex]f(x)=(\frac{2}{3} )^{x}[/tex]
Thus, the equation that matches the given table is [tex]f(x)=(\frac{2}{3} )^{x}[/tex]
