Question

The table below represents a geometric sequence.

1 4
2 20
3 100

Determine the recursive function that defines the sequence.

Answer 1

To determine the recursive function that defines the given geometric sequence, we first need to identify the first term and the common ratio.

The terms of the sequence are:
- \( a_1 = 4 \)
- \( a_2 = 20 \)
- \( a_3 = 100 \)

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio (\( r \)).

To find \( r \), we can calculate it using the ratio of consecutive terms:

\[
r = \frac{a_2}{a_1} = \frac{20}{4} = 5
\]

\[
r = \frac{a_3}{a_2} = \frac{100}{20} = 5
\]

This confirms that the common ratio \( r \) is indeed 5.

Now we can define the recursive function for the geometric sequence:

\[
a_n = r \cdot a_{n-1} \quad \text{for } n \geq 2
\]
where \( r = 5 \) and the initial condition \( a_1 = 4 \).

Thus, the recursive function is:

\[
a_n = 5 \cdot a_{n-1} \quad \text{for } n \geq 2
\]
with
\[
a_1 = 4
\]

This defines the given geometric sequence completely.