Answer:
Please check the explanation.
Step-by-step explanation:
Given the sequence
![40,\:10,\:\frac{5}{2},\:\frac{5}{8}](https://tex.z-dn.net/?f=40%2C%5C%3A10%2C%5C%3A%5Cfrac%7B5%7D%7B2%7D%2C%5C%3A%5Cfrac%7B5%7D%7B8%7D)
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=%5C%3Aa_n%3Da_0%5Ccdot%20r%5E%7Bn-1%7D)
Computing the ratios of all the adjacent terms
![\frac{10}{40}=\frac{1}{4},\:\quad \frac{\frac{5}{2}}{10}=\frac{1}{4},\:\quad \frac{\frac{5}{8}}{\frac{5}{2}}=\frac{1}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B40%7D%3D%5Cfrac%7B1%7D%7B4%7D%2C%5C%3A%5Cquad%20%5Cfrac%7B%5Cfrac%7B5%7D%7B2%7D%7D%7B10%7D%3D%5Cfrac%7B1%7D%7B4%7D%2C%5C%3A%5Cquad%20%5Cfrac%7B%5Cfrac%7B5%7D%7B8%7D%7D%7B%5Cfrac%7B5%7D%7B2%7D%7D%3D%5Cfrac%7B1%7D%7B4%7D)
The ratio of all the adjacent terms is the same and equal to
![r=\frac{1}{4}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B1%7D%7B4%7D)
Thus, the given sequence is a geometric sequence.
As the first element of the sequence is
![a_1=40](https://tex.z-dn.net/?f=a_1%3D40)
Therefore, the nth term is calculated as
![\:a_n=a_0\cdot r^{n-1}](https://tex.z-dn.net/?f=%5C%3Aa_n%3Da_0%5Ccdot%20r%5E%7Bn-1%7D)
![a_n=40\left(\frac{1}{4}\right)^{n-1}](https://tex.z-dn.net/?f=a_n%3D40%5Cleft%28%5Cfrac%7B1%7D%7B4%7D%5Cright%29%5E%7Bn-1%7D)
Put n = 5 to find the next term
![a_5=40\left(\frac{1}{4}\right)^{5-1}](https://tex.z-dn.net/?f=a_5%3D40%5Cleft%28%5Cfrac%7B1%7D%7B4%7D%5Cright%29%5E%7B5-1%7D)
![a_5=40\cdot \frac{1}{4^4}](https://tex.z-dn.net/?f=a_5%3D40%5Ccdot%20%5Cfrac%7B1%7D%7B4%5E4%7D)
![a_5=\frac{40}{4^4}](https://tex.z-dn.net/?f=a_5%3D%5Cfrac%7B40%7D%7B4%5E4%7D)
![=\frac{2^3\cdot \:5}{2^8}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%5E3%5Ccdot%20%5C%3A5%7D%7B2%5E8%7D)
![a_5=\frac{5}{2^5}](https://tex.z-dn.net/?f=a_5%3D%5Cfrac%7B5%7D%7B2%5E5%7D)
![a_5=\frac{5}{32}](https://tex.z-dn.net/?f=a_5%3D%5Cfrac%7B5%7D%7B32%7D)
now, Put n = 6 to find the 6th term
![a_6=40\left(\frac{1}{4}\right)^{6-1}](https://tex.z-dn.net/?f=a_6%3D40%5Cleft%28%5Cfrac%7B1%7D%7B4%7D%5Cright%29%5E%7B6-1%7D)
![a_6=40\cdot \frac{1}{4^5}](https://tex.z-dn.net/?f=a_6%3D40%5Ccdot%20%5Cfrac%7B1%7D%7B4%5E5%7D)
![a_6=\frac{40}{4^5}](https://tex.z-dn.net/?f=a_6%3D%5Cfrac%7B40%7D%7B4%5E5%7D)
![=\frac{2^3\cdot \:5}{2^{10}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B2%5E3%5Ccdot%20%5C%3A5%7D%7B2%5E%7B10%7D%7D)
![a_6=\frac{5}{2^7}](https://tex.z-dn.net/?f=a_6%3D%5Cfrac%7B5%7D%7B2%5E7%7D)
![a_6=\frac{5}{128}](https://tex.z-dn.net/?f=a_6%3D%5Cfrac%7B5%7D%7B128%7D)
Thus, the next two terms of the sequence 40, 10, 5/2, 5/8... is: