Respuesta :

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.




 
                         

Answer:

The 6th term in the geometric sequence is either 18750 or -18750.

Step-by-step explanation:

Given information: In the given GP

[tex]t_3=150[/tex]

[tex]t_5=3750[/tex]

The nth term of a GP is

[tex]t_n=ar^{n-1}[/tex]

where, a is first term and r is common ratio.

Third term of the GP is 150, so

[tex]t_3=ar^{3-1}[/tex]

[tex]150=ar^2[/tex]          .... (1)

Fifth term of the GP is 3750, so

[tex]t_5=ar^{5-1}[/tex]

[tex]3750=ar^4[/tex]          .... (2)

Divide equation (2) by equation (1).

[tex]\frac{3750}{150}=\frac{ar^4}{ar^2}[/tex]

[tex]25=r^2[/tex]

[tex]\pm 5=r[/tex]

The value of common ratio is either 5 or -5.

Put the value of r² in equation (1).

[tex]150=a(25)[/tex]

[tex]6=a[/tex]

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

[tex]t_6=ar^{6-1}=ar^5=6(5)^5=18750[/tex]

If the first term of GP is 6 and common difference is -5, then 6th term is

[tex]t_6=ar^{6-1}=ar^5=6(-5)^5=-18750[/tex]

Therefore the 6th term in the geometric sequence is either 18750 or -18750.

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