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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. $\newline$ $6, \, -18, \, 54, \, \dots$ $\newline$ 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

6, \, -18, \, 54, \, \dots

\newline

Write your answer using decimals and integers.

\newline

a_n =

\_\_\_\_

(

\_\_\_\_

)

^{n - 1}

Identify sequence type: Identify the type of sequence. $\newline$ We have the sequence: $6, -18, 54, \ldots$ $\newline$ To determine if the sequence is arithmetic or geometric, we look at the relationship between consecutive terms. $\newline$ $6$ to $-18$ is multiplied by $-3$ , and $-18$ to $54$ is also multiplied by $-3$ . $\newline$ Since each term is multiplied by a common ratio, the sequence is geometric.
Determine first term and ratio: Determine the first term ( $a_1$ ) and the common ratio ( $r$ ). $\newline$ The first term: $a_1 = 6$ $\newline$ To find the common ratio, divide the second term by the first term: $r = \frac{-18}{6} = -3$
Write nth term formula: Write the formula for the nth term of a geometric sequence. $\newline$ The general formula for the nth term of a geometric sequence is: $\newline$ $a_{n} = a_{1} \times r^{(n - 1)}$
Substitute values into formula: Substitute the values of $a_1$ and $r$ into the formula. $\newline$ Substitute $6$ for $a_1$ and $-3$ for $r$ into the formula: $\newline$ $a_n = 6 \times (-3)^{n - 1}$

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