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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. $\newline$ $–1,\,4,\,–16,\,\dots$ $\newline$ 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

–1,\,4,\,–16,\,\dots

\newline

Write your answer using decimals and integers.

\newline

a_n =

\_\_\_\_

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

Identify Pattern: Identify the pattern in the sequence. $\newline$ The given sequence is $-1, 4, -16, \ldots$ $\newline$ We notice that each term changes sign and is multiplied by a common ratio. $\newline$ This indicates that the sequence is geometric with alternating signs.
Determine Terms: Determine the first term $a_{1}$ and the common ratio $r$ . The first term is $a_{1} = -1$ . To find the common ratio, we divide the second term by the first term: $r = \frac{4}{(-1)} = -4$ .
Write Formula: Write the general formula for the $n$ th term of a geometric sequence. $\newline$ The $n$ th term of a geometric sequence is given by $a_n = a_1 \cdot r^{(n - 1)}$ .
Substitute Values: Substitute the values of $a_1$ and $r$ into the formula. $\newline$ We have $a_1 = -1$ and $r = -4$ . $\newline$ So, the formula for the nth term is $a_n = (-1) \times (-4)^{(n - 1)}$ .

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