Answer:
The nth term for the geometric sequence is given by:
![a_n = a_1 \cdot r^{n-1}](https://tex.z-dn.net/?f=a_n%20%3D%20a_1%20%5Ccdot%20r%5E%7Bn-1%7D)
where,
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
and ![r = -3](https://tex.z-dn.net/?f=r%20%3D%20-3)
Since,
,
ans so on...
Substitute this in [1] we have;
![a_n = -4 \cdot (-3)^{n-1}](https://tex.z-dn.net/?f=a_n%20%3D%20-4%20%5Ccdot%20%28-3%29%5E%7Bn-1%7D)
Substitute n = 5 we have;
![a_5 = -4 \cdot (-3)^{4}](https://tex.z-dn.net/?f=a_5%20%3D%20-4%20%5Ccdot%20%28-3%29%5E%7B4%7D)
⇒![a_5 = -4 \cdot 81 = -324](https://tex.z-dn.net/?f=a_5%20%3D%20-4%20%5Ccdot%2081%20%3D%20-324)
Recursive formula for the geometric sequence is given by:
![a_n = r \cdot a_{n-1}](https://tex.z-dn.net/?f=a_n%20%3D%20r%20%5Ccdot%20a_%7Bn-1%7D)
Substitute the given values we have;
![a_n = -3 \cdot a_{n-1}](https://tex.z-dn.net/?f=a_n%20%3D%20-3%20%5Ccdot%20a_%7Bn-1%7D)
Therefore, the recursive formula and the 5th term in the sequence is,
; ![a_5 = -324](https://tex.z-dn.net/?f=a_5%20%3D%20-324)