Question

write an explicit formula that represents the sequence d following recursive formula:
$a_1 = 3$ and $a_n = -2a_{n - 1}$

Explanation:

Step1: Identify Sequence Type

The recursive formula \( a_1 = 3 \) and \( a_n=-2a_{n - 1} \) shows each term is -2 times the previous term, so it's a geometric sequence. For a geometric sequence, the first term \( a_1 = 3 \) and common ratio \( r=-2 \).

Step2: Recall Geometric Explicit Formula

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

Step3: Substitute Values

Substitute \( a_1 = 3 \) and \( r=-2 \) into the formula: \( a_n = 3\cdot(-2)^{n - 1} \).

Answer:

\( a_n = 3(-2)^{n - 1} \)