Question

Find the 6th term in the geometric sequence with t3 = 150 and t5 = 3750.

Answer 1

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.

Answer 2

Determine first the common ratio (r) of the geometric sequence by,
 
                                r = (t5 / t3)^(1/(5 - 3))

Substituting the known values from the given,
 
                                r = (3750 / 150)^(1/2) = 5

The sixth term may be obtained by multiplying the fifth term (t5 = 3750) by the common ratio. This is shown below,
 
                                   t6 = (3750) x 5 = 18750

Thus, the 6th term of the geometric sequence is 18750.