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, 10, 20, ...
Write your answer using decimals and integers.
an = ____ (___)^n-1

Answer 1

The sequence you've provided is \( 5, 10, 20, \ldots \).

First, let's identify the pattern of the sequence:
- The first term \( a_1 = 5 \)
- The second term \( a_2 = 10 \)
- The third term \( a_3 = 20 \)

We can see that each term is obtained by multiplying the previous term by 2:
- From \( 5 \) to \( 10 \): \( 5 \times 2 = 10 \)
- From \( 10 \) to \( 20 \): \( 10 \times 2 = 20 \)

This suggests that the sequence is geometric with a first term of \( 5 \) and a common ratio of \( 2 \).

The general formula for the n-th term of a geometric sequence can be expressed as:
\[
a_n = a_1 \times r^{n-1}
\]
where \( a_1 \) is the first term and \( r \) is the common ratio.

In this case, \( a_1 = 5 \) and \( r = 2 \).

Thus, the equation for the sequence is:
\[
a_n = 5 \times (2)^{n-1}
\]

So, your answer is:
\[
a_n = 5 (2)^{n-1}
\]