Question

write a recursive formula for the following geometric sequence.
{1, -3, 9, -27, ...}

select one:
a. $a_n = -3a_{n - 1}$, for $n \geq 2$, $a_1 = 1$
b. $a_n = 3a_{n - 1}$, for $n \geq 2$, $a_1 = 1$
c. $a_n = -\frac{1}{3}a_{n - 1}$, for $n \geq 2$, $a_1 = 1$
d. $a_n = \frac{1}{3}a_{n - 1}$, for $n \geq 2$, $a_1 = 1$

Explanation:

Step1: Identify common ratio

Divide term by prior term: $\frac{-3}{1}=-3$, $\frac{9}{-3}=-3$, $\frac{-27}{9}=-3$

Step2: Define recursive formula

Recursive rule: $a_n = r \cdot a_{n-1}$, with $a_1=1$
Substitute $r=-3$: $a_n = -3a_{n-1}$ for $n\geq2$

Step3: Match to options

The formula matches option A.

Answer:

A. $a_n = -3a_{n-1}$, for $n \geq 2$, $a_1 = 1$