Answer:
An explicit rule for the nth term of the sequence will be:
Thus, option (A) is true.
Step-by-step explanation:
Given the sequence
![-4, -8, -16, -32, ...](https://tex.z-dn.net/?f=-4%2C%20-8%2C%20-16%2C%20-32%2C%20...)
A geometric sequence has a constant ratio r and is defined by
![a_n=a_0\cdot r^{n-1}](https://tex.z-dn.net/?f=a_n%3Da_0%5Ccdot%20r%5E%7Bn-1%7D)
Computing the ratios of all the adjacent terms
![\frac{-8}{-4}=2,\:\quad \frac{-16}{-8}=2,\:\quad \frac{-32}{-16}=2](https://tex.z-dn.net/?f=%5Cfrac%7B-8%7D%7B-4%7D%3D2%2C%5C%3A%5Cquad%20%5Cfrac%7B-16%7D%7B-8%7D%3D2%2C%5C%3A%5Cquad%20%5Cfrac%7B-32%7D%7B-16%7D%3D2)
As the ratio 'r' is the same.
so
![r=2](https://tex.z-dn.net/?f=r%3D2)
as
![a_1=-4](https://tex.z-dn.net/?f=a_1%3D-4)
Hence, the nth term of the sequence will be:
![a_n=a_0\cdot r^{n-1}](https://tex.z-dn.net/?f=a_n%3Da_0%5Ccdot%20r%5E%7Bn-1%7D)
substituting the values
and ![a_1=-4](https://tex.z-dn.net/?f=a_1%3D-4)
Therefore, an explicit rule for the nth term of the sequence will be:
Thus, option (A) is true.