Question

Mark has a garden and likes to put flowers from his garden into pretty vases. Mark put 2 flowers in the first vase, 6 flowers in the second vase, 18 flowers in the third vase, and 54 flowers in the fourth vase. What kind of sequence is this?

Answer 1

To determine the type of sequence, let's analyze the number of flowers in each vase:

- First vase: 2 flowers
- Second vase: 6 flowers
- Third vase: 18 flowers
- Fourth vase: 54 flowers

We can observe the relationship between the quantities in consecutive vases:

- From the 1st vase to the 2nd vase: \(6 = 2 \times 3\)
- From the 2nd vase to the 3rd vase: \(18 = 6 \times 3\)
- From the 3rd vase to the 4th vase: \(54 = 18 \times 3\)

Each term is obtained by multiplying the previous term by 3. This indicates that the sequence is a geometric sequence where the first term is \(2\) and the common ratio is \(3\).

The general form for the \(n\)-th term of a geometric sequence can be written as:

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

where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

In this case:
- \(a_1 = 2\),
- \(r = 3\).

Thus, the \(n\)-th term of this sequence can be expressed as:

\[
a_n = 2 \cdot 3^{(n-1)}
\]

In conclusion, the sequence of flowers Mark placed in the vases is a geometric sequence with a common ratio of 3.