Question

Find an explicit formula for the arithmetic sequence
\[81,54,27,0,\unicode{0x2026}\].
Note: the first term should be
\[\text{\it{a(1)}}\].
\[a(n)=\]

Answer 1

To find an explicit formula for the arithmetic sequence \( 81, 54, 27, 0, \ldots \), we need to identify the first term and the common difference of the sequence.

1. **Identify the first term**:
The first term \( a(1) \) is \( 81 \).

2. **Calculate the common difference**:
The common difference \( d \) is calculated by subtracting any term from the subsequent term.

\[
d = a(2) - a(1) = 54 - 81 = -27
\]
\[
d = a(3) - a(2) = 27 - 54 = -27
\]
\[
d = a(4) - a(3) = 0 - 27 = -27
\]

The common difference \( d \) is consistent and equals \( -27 \).

3. **General formula for the nth term**:
The formula for the nth term of an arithmetic sequence is given by:
\[
a(n) = a(1) + (n - 1) \cdot d
\]
Substituting \( a(1) = 81 \) and \( d = -27 \):
\[
a(n) = 81 + (n - 1)(-27)
\]
This simplifies to:
\[
a(n) = 81 - 27(n - 1)
\]
Expanding it further:
\[
a(n) = 81 - 27n + 27 = 108 - 27n
\]

Thus, the explicit formula for the arithmetic sequence is:
\[
\boxed{a(n) = 108 - 27n}
\]