Question

use the explicit formula to find a recursive formula for the sequence $a_n$. write your answer in simplest form.
the recursive formula should depend on $a_{n - 1}$.
$a_n = -50(-5)^{n - 1}$
$a_n = \boxed{}$

Explanation:

Step1: Recall recursive formula for geometric sequence

For a geometric sequence, the recursive formula is \( a_n = r \cdot a_{n - 1} \), where \( r \) is the common ratio. We need to find the common ratio from the explicit formula \( a_n=-50(-5)^{n - 1} \).

Step2: Identify the common ratio

The explicit formula for a geometric sequence is \( a_n = a_1r^{n - 1} \), where \( a_1 \) is the first term and \( r \) is the common ratio. Comparing \( a_n=-50(-5)^{n - 1} \) with \( a_n = a_1r^{n - 1} \), we can see that \( a_1=-50 \) and \( r = - 5 \).

Step3: Write the recursive formula

Using the recursive formula for a geometric sequence \( a_n=r\cdot a_{n - 1} \) with \( r=-5 \), we get \( a_n=-5\cdot a_{n - 1} \). We also need to consider the initial term. For \( n = 1 \), \( a_1=-50(-5)^{1 - 1}=-50(1)=-50 \). But the problem asks for the recursive formula depending on \( a_{n - 1} \), so the recursive formula is \( a_n=-5a_{n - 1} \) with \( a_1 = - 50 \). But since we only need the part depending on \( a_{n-1} \) for the formula (assuming the initial term is already considered in the sequence definition), the formula for \( a_n \) in terms of \( a_{n - 1} \) is \( a_n=-5a_{n - 1} \).

Answer:

\( a_n=-5a_{n - 1} \) (with \( a_1=-50 \), but the formula for \( a_n \) in terms of \( a_{n-1} \) is \( -5a_{n - 1} \))