Question

Write an explicit formula for the following recursive formula.
a Subscript nequalsa Subscript n minus 1plus8​; a 1equals1
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Part 1
a Subscript nequals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

The given recursive formula is:

\[
a_n = a_{n-1} + 8 \quad \text{with} \quad a_1 = 1
\]

To derive the explicit formula, we can start from the base case and look for a pattern.

1. **Base case**: \( a_1 = 1 \)
2. **Next term**:
\[
a_2 = a_1 + 8 = 1 + 8 = 9
\]
3. **Next term**:
\[
a_3 = a_2 + 8 = 9 + 8 = 17
\]
4. **Next term**:
\[
a_4 = a_3 + 8 = 17 + 8 = 25
\]

From these calculations, we can see a pattern emerging:

- \( a_1 = 1 \)
- \( a_2 = 1 + 8 \)
- \( a_3 = 1 + 8 + 8 = 1 + 2 \times 8 \)
- \( a_4 = 1 + 3 \times 8 \)

It appears we can express \( a_n \) in terms of the number of times we add \( 8 \). Specifically, we add \( 8 \) a total of \( n - 1 \) times:

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

Now let's simplify that expression:

\[
a_n = 1 + 8(n - 1) = 1 + 8n - 8 = 8n - 7
\]

So the explicit formula for \( a_n \) is:

\[
\boxed{8n - 7}
\]