Answer:
The 6th term in the geometric sequence is either 18750 or -18750.
Step-by-step explanation:
Given information: In the given GP
\(t_3=150\)
\(t_5=3750\)
The nth term of a GP is
\(t_n=ar^{n-1}\)
where, a is first term and r is common ratio.
Third term of the GP is 150, so
\(t_3=ar^{3-1}\)
\(150=ar^2\) .... (1)
Fifth term of the GP is 3750, so
\(t_5=ar^{5-1}\)
\(3750=ar^4\) .... (2)
Divide equation (2) by equation (1).
\(\frac{3750}{150}=\frac{ar^4}{ar^2}\)
\(25=r^2\)
\(\pm 5=r\)
The value of common ratio is either 5 or -5.
Put the value of r² in equation (1).
\(150=a(25)\)
\(6=a\)
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
\(t_6=ar^{6-1}=ar^5=6(5)^5=18750\)
If the first term of GP is 6 and common difference is -5, then 6th term is
\(t_6=ar^{6-1}=ar^5=6(-5)^5=-18750\)
Therefore the 6th term in the geometric sequence is either 18750 or -18750.
