To find the equation describing the sequence \(5, 20, 80,\) we first observe the pattern in the terms.
- The first term \(a_1 = 5\)
- The second term \(a_2 = 20\)
- The third term \(a_3 = 80\)
Next, we can see how each term relates to the previous term:
- From \(5\) to \(20\), we multiply by \(4\).
- From \(20\) to \(80\), we also multiply by \(4\).
We can notice that this sequence is a geometric sequence where each term is multiplied by \(4\):
- \(a_2 = a_1 \cdot 4\)
- \(a_3 = a_2 \cdot 4\)
In the general formula for a geometric sequence, the \(n\)-th term is given by:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term's position.
In this case:
- \(a_1 = 5\)
- \(r = 4\)
Now we can write the formula:
\[ a_n = 5 \cdot 4^{(n-1)} \]
Thus, the final equation describing the sequence is:
\[ a_n = 5(4)^{(n-1)} \]