Answer:
The 6th term in the geometric sequence is either 18750 or -18750.
Step-by-step explanation:
Given information: In the given GP
![t_3=150](https://tex.z-dn.net/?f=t_3%3D150)
![t_5=3750](https://tex.z-dn.net/?f=t_5%3D3750)
The nth term of a GP is
![t_n=ar^{n-1}](https://tex.z-dn.net/?f=t_n%3Dar%5E%7Bn-1%7D)
where, a is first term and r is common ratio.
Third term of the GP is 150, so
![t_3=ar^{3-1}](https://tex.z-dn.net/?f=t_3%3Dar%5E%7B3-1%7D)
.... (1)
Fifth term of the GP is 3750, so
![t_5=ar^{5-1}](https://tex.z-dn.net/?f=t_5%3Dar%5E%7B5-1%7D)
.... (2)
Divide equation (2) by equation (1).
![\frac{3750}{150}=\frac{ar^4}{ar^2}](https://tex.z-dn.net/?f=%5Cfrac%7B3750%7D%7B150%7D%3D%5Cfrac%7Bar%5E4%7D%7Bar%5E2%7D)
![25=r^2](https://tex.z-dn.net/?f=25%3Dr%5E2)
![\pm 5=r](https://tex.z-dn.net/?f=%5Cpm%205%3Dr)
The value of common ratio is either 5 or -5.
Put the value of r² in equation (1).
![150=a(25)](https://tex.z-dn.net/?f=150%3Da%2825%29)
![6=a](https://tex.z-dn.net/?f=6%3Da)
The first term of the GP is 6.
If the first term of GP is 6 and common difference is 5, then 6th term is
![t_6=ar^{6-1}=ar^5=6(5)^5=18750](https://tex.z-dn.net/?f=t_6%3Dar%5E%7B6-1%7D%3Dar%5E5%3D6%285%29%5E5%3D18750)
If the first term of GP is 6 and common difference is -5, then 6th term is
![t_6=ar^{6-1}=ar^5=6(-5)^5=-18750](https://tex.z-dn.net/?f=t_6%3Dar%5E%7B6-1%7D%3Dar%5E5%3D6%28-5%29%5E5%3D-18750)
Therefore the 6th term in the geometric sequence is either 18750 or -18750.