Answer:
The sequence is geometric
The recursive formula for the sequence is:
![A_n=\cdot A_{n-1}\cdot -3, A_1=-2](https://tex.z-dn.net/?f=A_n%3D%5Ccdot%20A_%7Bn-1%7D%5Ccdot%20-3%2C%20A_1%3D-2)
The explicit formula for the sequence is
![A_n= - 2 {( - 3)}^{n - 1}](https://tex.z-dn.net/?f=A_n%3D%20-%202%20%7B%28%20-%203%29%7D%5E%7Bn%20-%201%7D%20)
Explanation:
The given sequence is -2, 6, -18,54
There is a common ratio of
![r = \frac{6}{ - 2} = \frac{ - 18}{6} = \frac{54}{ - 18} = - 3](https://tex.z-dn.net/?f=r%20%3D%20%20%5Cfrac%7B6%7D%7B%20-%202%7D%20%20%3D%20%20%5Cfrac%7B%20-%2018%7D%7B6%7D%20%20%3D%20%20%5Cfrac%7B54%7D%7B%20-%2018%7D%20%20%3D%20%20-%203)
Since there is a common ratio, the sequence is geometric.
The first term of this sequence is
![A_1=-2](https://tex.z-dn.net/?f=A_1%3D-2)
The recursive definition is given by:
![A_n=r\cdot A_{n-1}](https://tex.z-dn.net/?f=A_n%3Dr%5Ccdot%20A_%7Bn-1%7D)
We substitute r=-3 to get,
![A_n= - 3\cdot A_{n-1}](https://tex.z-dn.net/?f=A_n%3D%20-%203%5Ccdot%20A_%7Bn-1%7D)
The explicit formula is
![A_n= - 2 {( - 3)}^{n - 1}](https://tex.z-dn.net/?f=A_n%3D%20-%202%20%7B%28%20-%203%29%7D%5E%7Bn%20-%201%7D%20)