Question

whats the explicit rule for the sequence 3, -6, 12, -24, 48, ...?
a) (a_n = 2(-3)^{n - 1})
b) (a_n = 3(-2)^{n - 1})
c) (a_n = 3(-2)^n)
d) (a_n = 3(2)^{n - 1})
question 10 (5 points)
given the recursive formula shown, what are the first 4 terms of the sequence? (f(n)=\begin{cases}f(1)=5 \text{ if } n = 1\\f(n)=4f(n - 1)\text{ if }n>1end{cases})
a) 5, 20, 80, 320
b) 5, 25, 125, 625
c) 5, 14, 60, 236

Explanation:

Step1: Identify the type of sequence

The given sequence 3, -6, 12, -24, 48, ... is a geometric sequence since there is a common - ratio between consecutive terms. The common - ratio $r=\frac{-6}{3}=- 2$. The general form of the explicit rule for a geometric sequence is $a_{n}=a_{1}r^{n - 1}$, where $a_{1}$ is the first term and $r$ is the common ratio. Here, $a_{1}=3$ and $r=-2$. So the explicit rule is $a_{n}=3(-2)^{n - 1}$.

Step2: Analyze the recursive formula for the second question

For the recursive formula $f(n)=

$$\begin{cases}f(1) = 5, &n = 1\\f(n)=4f(n - 1),&n>1\end{cases}$$

$
When $n = 1$, $f(1)=5$.
When $n = 2$, $f(2)=4f(1)=4\times5 = 20$.
When $n = 3$, $f(3)=4f(2)=4\times20 = 80$.
When $n = 4$, $f(4)=4f(3)=4\times80 = 320$.

Answer:

For the first question: B. $a_{n}=3(-2)^{n - 1}$
For the second question: A. 5, 20, 80, 320