Question

Write a recursive formula for the following sequence and also identify the seventh them. You are welcome to submit an image of handwritten work. If you choose to type then use the following notation to indicate terms; a_n and a_(n-1). To earn full credit be sure to share all work/calculations and thinking.a_n = {-2, \frac{2}{3}, -\frac{2}{9}, \frac{2}{27}}

Answer 1

Step-by-step explanation:

We are given a sequence and the first and second terms are;

`\begin{gathered} a_1=-2 \\ a_2=(2)/(3) \end{gathered}`

The common ratio is determined by dividing a term by its preceeding term. Hence, we can derive the common ratio by dividing the second term by the first.

`\begin{gathered} Common\text{ }ratio: \\ r=(2)/(3)/(-2)/(1) \end{gathered}`
`r=(2)/(3)*(1)/(-2)`
`r=-(1)/(3)`

The recursive formula for this sequence would be given as follows;

`a_n=a_(n-1)* r`

Where the variables are;

`\begin{gathered} a_n=nth\text{ term} \\ r=-(1)/(3) \end{gathered}`

We now have;

`a_n=a_(n-1)(-(1)/(3))`

For the 5th term we would have;

`a_5=a_4(-(1)/(3))`
`a_5=(2)/(27)(-(1)/(3))`
`a_5=-(2)/(81)`

For the 6th term we would have;

`a_6=a_5(-(1)/(3))`
`a_6=-(2)/(81)*(-(1)/(3))`
`a_6=(2)/(243)`

For the 7th term we would have;

`a_7=a_6(-(1)/(3))`
`a_7=(2)/(243)(-(1)/(3))`
`a_7=-(2)/(729)`

ANSWER:

`\begin{gathered} Recursive\text{ }formula: \\ a_n=a_(n-1)(-(1)/(3)) \\ 7th\text{ }term: \\ a_7=-(2)/(729) \end{gathered}`