Question

Write the explicit formula for the geometric sequence represented by the recursive formula.

a1 = 4
an = 5 · an-1

Answer 1

To convert the recursive formula of the geometric sequence into an explicit formula, we start by observing the sequence defined by the given recursive relations:

1. The first term \( a_1 = 4 \).
2. The recursive relation is \( a_n = 5 \cdot a_{n-1} \).

This shows that each term is obtained by multiplying the previous term by \( 5 \). Therefore, we can write out the first few terms to identify the pattern:

- \( a_1 = 4 \)
- \( a_2 = 5 \cdot a_1 = 5 \cdot 4 = 20 \)
- \( a_3 = 5 \cdot a_2 = 5 \cdot 20 = 100 \)
- \( a_4 = 5 \cdot a_3 = 5 \cdot 100 = 500 \)

Now we can see that the terms can be expressed in a general form. For any term \( a_n \), we can express it as:

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

where \( a_1 = 4 \) is the first term, and \( r = 5 \) is the common ratio.

Thus, the explicit formula for the sequence can be written as:

\[
a_n = 4 \cdot 5^{n-1}
\]

This formula allows us to calculate any term in the sequence \( a_n \) directly without needing to compute all previous terms.