A geometric sequence is defined recursively by an = 6an -1 . The first term of the sequence is 0.75. Which of the following is the explicit formula for the nth term of the sequence?

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[tex] u_{1}= \frac{3}{4} \\ u_{2}= \frac{3}{4}*6^{1} \\ u_{3}= \frac{3}{4}*6^{2} \\ u_{4}= \frac{3}{4}*6^{3} \\ ...\\ \boxed{u_{n}= \dfrac{3}{4}*6^{n-1} } \\ [/tex]

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DelcieRiveria DelcieRiveria · Answer 2 · 2019-06-04T13:39:05+02:00

Answer:

The explicit formula for the nth term of the sequence is [tex]a_n=0.75(6)^{n-1}[/tex].

Step-by-step explanation:

The recursive formula of a GP is

[tex]a_n=6a_{n-1}[/tex]

It is given that the first term is 0.75, it means

[tex]a_1=0.75[/tex]

The next terms of GP are

[tex]a_2=6a_{2-1}=6\times (0.75)=4.5[/tex]

[tex]a_3=6a_{3-1}=6\times (4.5)=27[/tex]

The GP is defined as

[tex]0.75,4.5,27[/tex]

Here the first term is 0.75 and the common ratio is 6.

The explicit formula for the nth term of a GP is

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

The explicit formula for the nth term of the sequence

[tex]a_n=0.75(6)^{n-1}[/tex]

Therefore the explicit formula for the nth term of the sequence is [tex]a_n=0.75(6)^{n-1}[/tex].