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Write an equation to describe the sequence below. Use $n$ to represent the position of a term in the sequence, where $n = 1$ for the first term. $\newline$ $–5, \, –15, \, –45, \, \dots$ $\newline$ Write your answer using decimals and integers. $\newline$ $a_n =$ $\_\_\_\_$ ( $\_\_\_\_$ ) $^{n - 1}$

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Write variable expressions for geometric sequences

Full solution

Q. Write an equation to describe the sequence below. Use

n

to represent the position of a term in the sequence, where

n = 1

for the first term.

\newline

–5, \, –15, \, –45, \, \dots

\newline

Write your answer using decimals and integers.

\newline

a_n =

\_\_\_\_

(

\_\_\_\_

)

^{n - 1}

Sequence Type: We have: $\newline$ – $5$ , – $15$ , – $45$ , ... $\newline$ Is the given sequence geometric or arithmetic? $\newline$ – $5$ , – $15$ , – $45$ , ... $\newline$ Here, each term is multiplied by a common ratio. $\newline$ The given sequence is geometric.
Determine Values: $–5, –15, –45, \ldots$ $\newline$ Determine the values of $a_1$ and $r$ of the sequence. $\newline$ First term: $a_1 = –5$ $\newline$ Common ratio: $r = \frac{–15}{–5} = 3$
Expression for $a_n$ : We have: $\newline$ $a_1 = -5$ $\newline$ $r = 3$ $\newline$ Determine an expression to describe $a_n$ . $\newline$ Substitute $-5$ for $a_1$ and $3$ for $r$ . $\newline$ $a_n = a_1 (r)^{n - 1}$ $\newline$ $a_n = -5(3)^{n-1}$

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