Question

For the following geometric sequence find the recursive formula and the 5th term in the sequence. In your final answer, include all of your work. {-4, 12, -36, ...}

Answer 1

Each term is -3 times of the previous term

$f(n)=-3f(n-1)$ is the recursive sequence because math

5th term is 4th term times -3 which is -3 times 3rd term (-36)
so -36 times -3 times -3=-324 is 5th term

Answer 2

Answer:

The nth term for the geometric sequence is given by:

$a_n = a_1 \cdot r^{n-1}$

where,

$a_1$ is the first term

r is the common ratio term

n is the number of terms.

Given the sequence:

-4, 12, -36,......

This is a geometric sequence with  $a_1 = -4$ and  $r = -3$

Since,

$\frac{12}{-4} = -3$,

$\frac{-36}{12} = -3$ ans so on...

Substitute this in [1] we have;

$a_n = -4 \cdot (-3)^{n-1}$

Substitute n = 5 we have;

$a_5 = -4 \cdot (-3)^{4}$

⇒$a_5 = -4 \cdot 81 = -324$

Recursive formula for the geometric sequence is given by:

$a_n = r \cdot a_{n-1}$

Substitute the given values we have;

$a_n = -3 \cdot a_{n-1}$

Therefore, the recursive formula and the 5th term in the sequence is,

$a_n = -3 \cdot a_{n-1}$ ; $a_5 = -324$