Question

use the initial term and the recursive formula to find an explicit formula for the sequence $a_n$. write your answer in simplest form. $a_1 = 300$ $a_n = -5a_{n - 1}$ $a_n = \boxed{}$

Explanation:

Step1: Identify Sequence Type

The recursive formula \( a_n = -5a_{n - 1} \) and initial term \( a_1 = 300 \) indicate a geometric sequence. A geometric sequence has the form \( a_n = a_1 \cdot r^{n - 1} \), where \( r \) is the common ratio.

Step2: Determine Common Ratio

From \( a_n = -5a_{n - 1} \), the common ratio \( r = -5 \).

Step3: Write Explicit Formula

Substitute \( a_1 = 300 \) and \( r = -5 \) into the geometric sequence formula: \( a_n = 300 \cdot (-5)^{n - 1} \).

Answer:

\( a_n = 300(-5)^{n - 1} \)