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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. $\newline$ $9 , 18 , 36 , \ldots$ $\newline$ Write your answer using decimals and integers. $\newline$ $a_n =$ _(_) $^{n - 1}$

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Write variable expressions for geometric sequences

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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

9 , 18 , 36 , \ldots

\newline

Write your answer using decimals and integers.

\newline

a_n =

_____(_____)

^{n - 1}

Identify sequence type: Identify the type of sequence. $\newline$ The sequence is $9, 18, 36, \ldots$ To determine if it's arithmetic or geometric, we look at the pattern between the terms. The second term is twice the first term, and the third term is twice the second term. This pattern of multiplying by a constant means the sequence is geometric.
Find first term and common ratio: Find the first term and the common ratio. $\newline$ The first term $a_{1}$ is $9$ . To find the common ratio $r$ , we divide the second term by the first term: $r = \frac{18}{9} = 2$ . This ratio is consistent throughout the sequence, as each term is twice the previous term.
Write formula for nth term: Write the formula for the nth term of a geometric sequence. $\newline$ The general formula for the nth term of a geometric sequence is $a_n = a_1 \cdot r^{(n - 1)}$ . We will use this formula to describe the sequence.
Substitute values into formula: Substitute the values of $a_1$ and $r$ into the formula. $\newline$ We have $a_1 = 9$ and $r = 2$ . Substituting these values into the formula gives us $a_n = 9 \cdot 2^{(n - 1)}$ .
Verify formula with given terms: Verify the formula with the given terms. $\newline$ To ensure there are no mistakes, we can check the formula with the given terms. For $n = 1$ , $a_{n}$ should be $9$ : $a_{1} = 9 \times 2^{1 - 1} = 9 \times 2^{0} = 9 \times 1 = 9$ . For $n = 2$ , $a_{n}$ should be $18$ : $a_{2} = 9 \times 2^{2 - 1} = 9 \times 2^{1} = 9 \times 2 = 18$ . The formula works for the given terms.

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