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.
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