Answer:
Please check the explanation.
Step-by-step explanation:
Given the sequence
[tex]40,\:10,\:\frac{5}{2},\:\frac{5}{8}[/tex]
A geometric sequence has a constant ratio 'r' and is defined by
[tex]\:a_n=a_0\cdot r^{n-1}[/tex]
Computing the ratios of all the adjacent terms
[tex]\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}[/tex]
The ratio of all the adjacent terms is the same and equal to
[tex]r=\frac{1}{4}[/tex]
Thus, the given sequence is a geometric sequence.
As the first element of the sequence is
[tex]a_1=40[/tex]
Therefore, the nth term is calculated as
[tex]\:a_n=a_0\cdot r^{n-1}[/tex]
[tex]a_n=40\left(\frac{1}{4}\right)^{n-1}[/tex]
Put n = 5 to find the next term
[tex]a_5=40\left(\frac{1}{4}\right)^{5-1}[/tex]
[tex]a_5=40\cdot \frac{1}{4^4}[/tex]
[tex]a_5=\frac{40}{4^4}[/tex]
[tex]=\frac{2^3\cdot \:5}{2^8}[/tex]
[tex]a_5=\frac{5}{2^5}[/tex]
[tex]a_5=\frac{5}{32}[/tex]
now, Put n = 6 to find the 6th term
[tex]a_6=40\left(\frac{1}{4}\right)^{6-1}[/tex]
[tex]a_6=40\cdot \frac{1}{4^5}[/tex]
[tex]a_6=\frac{40}{4^5}[/tex]
[tex]=\frac{2^3\cdot \:5}{2^{10}}[/tex]
[tex]a_6=\frac{5}{2^7}[/tex]
[tex]a_6=\frac{5}{128}[/tex]
Thus, the next two terms of the sequence 40, 10, 5/2, 5/8... is:
