An explicit formula for a,, the nth term of the sequence 27, 9, 3, .... is [tex]a_n=(27) \times\left(\frac{1}{3}\right)^{n-1}[/tex].
What is Geometric Progression?
A geometric progression is a sequence in which every next term of the sequence is found by multiplying the previous term by a fixed ratio.
Any nth term of the sequence is found out the formula,
[tex]$a_n=a_1 \times r^{n-1}$[/tex]
where,
[tex]$a_n$[/tex] is the nth term,
[tex]$a_1$[/tex] is the first term,
r is the fixed common ratio.
Sequence, 27, 9, 3, 1 .
the first term, [tex]$a_1=27$[/tex],
As we can see from series 27,9,3, 1. the series is a geometric series And can be written as [tex]$3^3, 3^2, 3^1, 3^0$[/tex]. therefore, will follow the formula of a geometric series.
[tex]a_n=a_1 \times r^{n-1} \text {, }[/tex]
we know the value of r can be found using the formula,
[tex]r=\frac{a_n}{a_{n-1}}[/tex]
taking [tex]$\mathrm{n}=2$[/tex],
[tex]r=\frac{a_2}{a_{2-1}}=\frac{a_2}{a_1}=\frac{9}{27}=\frac{1}{3}[/tex]
Substituting the values in the formula of geometric progression we get,
[tex]$\begin{aligned}&a_n=a_1 \times r^{n-1} \\&a_n=27 \times \frac{1}{3}^{n-1} \\&a_n=(27) \times\left(\frac{1}{3}\right)^{n-1}\end{aligned}$[/tex]
Learn more about Substituting Geometric Progression:
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