Question

Write an equation to describe the sequence below.Use n to represent the position of a term in the sequence,where n = 1 for the first term.
5, 20, 80,
Write your answer using decimals and integers.
an = __ ( __ )^n-1

Answer 1

To find the equation describing the sequence \(5, 20, 80,\) we first observe the pattern in the terms.

1. The first term \(a_1 = 5\)
2. The second term \(a_2 = 20\)
3. 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:

\[
a_n = 5(4)^{(n-1)}
\]