Question

it says write an explicit and a recursive equation for the table bellow and it must work for each of the entries in the table

Answer 1

You use the next formula to get a explicit equation for a geometric sequence:

$a_n=a_1\cdot r^{n-1}$

In this case you use as the first data 2000 (term when n=1).

r is the common ratio between each term.

To find r you divide each term into the previous term as follow:

$\begin{gathered} \frac{2000}{1600}=\frac{5}{4} \\ \\ \frac{2500}{2000}=\frac{5}{4} \\ \\ \frac{3125}{2500}=\frac{5}{4} \\ \\ \frac{3906.25}{3125}=\frac{5}{4} \end{gathered}$

Then, you get the next explicit equation: writen in two different forms

$\begin{gathered} t_n=2000\cdot(\frac{5}{4})^{n-1} \\ \\ t_n=\frac{2000\cdot5^{n-1}}{4^{n-1}} \end{gathered}$

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For the recursive formula you have the next:

Where tn-1 is the previus term

For the given sequence:

$\begin{gathered} t_1=200 \\ \\ t_n=\frac{5}{4}(t_{n-1}) \end{gathered}$