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Find an explicit formula for the arithmetic sequence

81,54,27,0,dots". "
Note: the first term should be
a(1).

a(n)=

Find an explicit formula for the arithmetic sequence $\newline$ $81,54,27,0, \ldots \text {.. }$ $\newline$ Note: the first term should be $a$ ( $1$ ). $\newline$ a(n) = \(\square\)

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Write a formula for an arithmetic sequence

Full solution

Q. Find an explicit formula for the arithmetic sequence

\newline

81,54,27,0, \ldots \text {.. }

\newline

Note: the first term should be

a

(

1

).

\newline

a(n) = \(\square\)

Identify Sequence Type: Identify whether the given sequence is geometric or arithmetic. The sequence $81, 54, 27, 0, \ldots$ has a common difference between consecutive terms, so it is an arithmetic sequence.
Find First Term and Difference: Determine the first term ( $a_1$ ) and the common difference ( $d$ ) of the sequence. The first term $a_1$ is $81$ . To find the common difference, subtract the second term from the first term: $d = 54 - 81 = -27$ .
Use Explicit Formula: Use the explicit formula for an arithmetic sequence, $a_n = a_1 + (n-1)d$ , where $a_1$ is the first term and $d$ is the common difference. For this sequence, $a_1 = 81$ and $d = -27$ .
Substitute Values: Substitute the values of $a_{1}$ and $d$ into the formula to write an expression to describe the sequence. The expression for the sequence $81, 54, 27, 0, \ldots$ is $a_{n} = 81 + (n-1)(-27)$ .
Simplify Expression: Simplify the expression to find the explicit formula. $a_{n} = 81 - 27(n-1) = 81 - 27n + 27 = 108 - 27n$ .

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