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. $a_1 = -60$ $a_n = 3a_{n - 1}$ $a_n = \square$

Explanation:

Step1: Identify sequence type

This is a geometric sequence, since each term is 3 times the previous term, with first term $a_1=-60$ and common ratio $r=3$.

Step2: Recall geometric sequence formula

The explicit formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$.

Step3: Substitute values

Substitute $a_1=-60$ and $r=3$ into the formula:
$a_n = -60 \cdot 3^{n-1}$

Step4: Simplify the expression

We can rewrite $3^{n-1}$ as $\frac{3^n}{3}$, so:
$a_n = -60 \cdot \frac{3^n}{3} = -20 \cdot 3^n$
Alternatively, the form $a_n=-60 \cdot 3^{n-1}$ is also valid, but $a_n=-20 \cdot 3^n$ is fully simplified.

Answer:

$a_n = -20 \cdot 3^n$ (or equivalently $a_n = -60 \cdot 3^{n-1}$)