Answer:
possible values of 4th term is 80 & - 80
Step-by-step explanation:
The general term of a geometric series is given by
[tex]a(n)=ar^{n-1}[/tex]
Where a(n) is the nth term, r is the common ratio (a term divided by the term before it) and n is the number of term
- Given, 5th term is 40, we can write:
[tex]ar^{5-1}=40\\ar^4=40[/tex]
- Given, 7th term is 10, we can write:
[tex]ar^{7-1}=10\\ar^6=10[/tex]
We can solve for a in the first equation as:
[tex]ar^4=40\\a=\frac{40}{r^4}[/tex]
Now we can plug this into a of the 2nd equation:
[tex]ar^6=10\\(\frac{40}{r^4})r^6=10\\40r^2=10\\r^2=\frac{10}{40}\\r^2=\frac{1}{4}\\r=+-\sqrt{\frac{1}{4}} \\r=\frac{1}{2},-\frac{1}{2}[/tex]
Let's solve for a:
[tex]a=\frac{40}{r^4}\\a=\frac{40}{(\frac{1}{2})^4}\\a=640[/tex]
Now, using the general formula of a term, we know that 4th term is:
4th term = ar^3
Plugging in a = 640 and r = 1/2 and -1/2 respectively, we get 2 possible values of 4th term as:
[tex]ar^3\\1.(640)(\frac{1}{2})^3=80\\2.(640)(-\frac{1}{2})^3=-80[/tex]
possible values of 4th term is 80 & - 80
