Question

An arithmetic sequence has this recursive formula:
{a1=8
{an=an-1-6
What is the explicit formula for this sequence

Answer 1

To find the explicit formula for the arithmetic sequence defined by the recursive formula:

- \( a_1 = 8 \)
- \( a_n = a_{n-1} - 6 \)

we first note the following:

1. **Identifying the First Term:** The first term \( a_1 \) is given as 8.
2. **Finding the Common Difference:** The sequence decreases by 6 with each term, so the common difference \( d \) is \(-6\).

The general form of an arithmetic sequence can be expressed explicitly as:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Substituting the values we have:

- \( a_1 = 8 \)
- \( d = -6 \)

The explicit formula becomes:

\[
a_n = 8 + (n - 1)(-6)
\]

Now we can simplify this:

\[
a_n = 8 - 6(n - 1)
\]
\[
a_n = 8 - 6n + 6
\]
\[
a_n = 14 - 6n
\]

So, the explicit formula for the arithmetic sequence is:

\[
\boxed{14 - 6n}
\]