Question

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?

Answer 1

Hello,

$u_{1}= \frac{3}{4} \\ u_{2}= \frac{3}{4}*6^{1} \\ u_{3}= \frac{3}{4}*6^{2} \\ u_{4}= \frac{3}{4}*6^{3} \\ ...\\ \boxed{u_{n}= \dfrac{3}{4}*6^{n-1} }$

Answer 2

Answer:

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

Step-by-step explanation:

The recursive formula of a GP is

$a_n=6a_{n-1}$

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

$a_1=0.75$

The next terms of GP are

$a_2=6a_{2-1}=6\times (0.75)=4.5$

$a_3=6a_{3-1}=6\times (4.5)=27$

The GP is defined as

$0.75,4.5,27$

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

The explicit formula for the nth term of a GP is

$a_n=ar^{n-1}$

The explicit formula for the nth term of the sequence

$a_n=0.75(6)^{n-1}$

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