Question

What is the explicit rule for the geometric sequence?

500, 100, 20, 4, ...

Answer 1

Explanation:

By definition:

" Geometric progression - a sequence of numbers b₁, b₂, b₃ ... (members of the progression), in which each subsequent number, starting from the second, is obtained from the previous member by multiplying it by a certain number q (the denominator of the progression)"

bₙ = bₙ₋₁*q

q = bₙ / bₙ₋₁

For example:

q = b₃ / b₂ = 20 /100 = 1/5

Answer 2

nth term is

`\begin{equation*}% \mbox{\large\( %a_n = 500\cdot \left((1)/(5)\right)^(n-1)\)} %\end{equation*}\\\\`
`\begin{equation*}% \mbox{\large\( %a_n = 500\cdot ((1)/(5))^(n-1)\)} %\end{equation*}`
`a_n = 500 \cdot \left((1)/(5)\right)^(n-1)`

`\\\textrm {$a_n$ is the nth term $500$ is the first term and $(1)/(5)$ is the common ratio $(a_n)/(a_(n-1))$}`

Explanation:

For a geometric sequence, the rule is

`\begin{equation*}% \mbox{\large\( %a_n =a_1 * r^(n-1)\)} %\end{equation*}`
`a_n = a_(n-1) \cdot r^(n-1)`

where

`\\\textrm {$a_n$ is the nth term$a_1$ is the first termand$r$ is the common ratio $(a_n)/(a_(n-1))$}`

Here you can see that each term is one-fifth of the previous term

So

`r = (1)/(5)`

First term is 500

So nth term

`\begin{equation*}% \mbox{\large\( %a_n = 500\cdot \left((1)/(5)\right)^(n-1)\)} %\end{equation*}`
`a_n = 500 \cdot \left((1)/(5)\right)^(n-1)`