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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. $\newline$ $3,\,12,\,48,\,\ldots$ $\newline$ Write your answer using decimals and integers. $\newline$ $a_n =$ $\_\_\_\_$ $\left(\_\_\_\_\right)^{n - 1}$

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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

3,\,12,\,48,\,\ldots

\newline

Write your answer using decimals and integers.

\newline

a_n =

\_\_\_\_

\left(\_\_\_\_\right)^{n - 1}

Identify type of sequence: Identify the type of sequence. $\newline$ We have the sequence: $3, 12, 48, \ldots$ $\newline$ To determine if the sequence is arithmetic or geometric, we look at the relationship between consecutive terms. $\newline$ $3$ to $12$ is a multiplication by $4$ , and $12$ to $48$ is also a multiplication by $4$ . $\newline$ Since each term is multiplied by a common ratio to get the next term, the sequence is geometric.
Determine first term and ratio: Determine the first term ( $a_1$ ) and the common ratio ( $r$ ). $\newline$ The first term of the sequence is $a_1 = 3$ . $\newline$ To find the common ratio, we divide the second term by the first term: $r = \frac{12}{3} = 4$ .
Write formula for nth term: Write the formula for the nth term of a geometric sequence. $\newline$ The nth term of a geometric sequence is given by the formula $a_n = a_1 \cdot r^{(n - 1)}$ .
Substitute values into formula: Substitute the values of $a_1$ and $r$ into the formula. $\newline$ We have $a_1 = 3$ and $r = 4$ . $\newline$ Substituting these values into the formula gives us $a_n = 3 \times 4^{(n - 1)}$ .

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