Question

Anthony found some empty bins to put cans into for recycling. He put 2 empty cans in the first bin, 14 empty cans in the second bin, 98 empty cans in the third bin, and 686 empty cans in the fourth bin. What kind of sequence is this?

Answer 1

To determine the type of sequence created by the number of empty cans in each bin, let's list the number of cans in each bin:

1. First bin: \(2\)
2. Second bin: \(14\)
3. Third bin: \(98\)
4. Fourth bin: \(686\)

Next, we can label these quantities as follows:
- \(a_1 = 2\)
- \(a_2 = 14\)
- \(a_3 = 98\)
- \(a_4 = 686\)

Now, let's check how these numbers relate to one another. We can observe the ratio between each consecutive pair of terms:

\[
\frac{a_2}{a_1} = \frac{14}{2} = 7
\]
\[
\frac{a_3}{a_2} = \frac{98}{14} = 7
\]
\[
\frac{a_4}{a_3} = \frac{686}{98} = 7
\]

Since the ratio between each consecutive term is constant (specifically, it is \(7\)), this indicates that the sequence is a geometric sequence with a common ratio of \(7\).

To write a general formula for the \(n\)-th term of a geometric sequence, we use the formula:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where \(a_1\) is the first term and \(r\) is the common ratio. In this case:

- First term \(a_1 = 2\)
- Common ratio \(r = 7\)

Thus, the general term for the sequence can be expressed as:

\[
a_n = 2 \cdot 7^{(n-1)}
\]

In conclusion, the sequence Anthony created is a **geometric sequence** with a common ratio of \(7\).