Question

Use the initial term and the recursive formula to find an explicit formula for the sequence an. Write your answer in simplest form.
a1
= 15
an
= an–1+16
an=

Answer 1

To find an explicit formula for the sequence defined by the initial term \( a_1 = 15 \) and the recursive formula \( a_n = a_{n-1} + 16 \), we can analyze how the sequence behaves.

### Step 1: Calculate the first few terms
Let's compute the first few terms of the sequence:

- \( a_1 = 15 \)
- \( a_2 = a_1 + 16 = 15 + 16 = 31 \)
- \( a_3 = a_2 + 16 = 31 + 16 = 47 \)
- \( a_4 = a_3 + 16 = 47 + 16 = 63 \)

From these calculations, we observe:

\[
\begin{align*}
a_1 & = 15 \\
a_2 & = 31 \\
a_3 & = 47 \\
a_4 & = 63 \\
\end{align*}
\]

### Step 2: Identify a pattern
The difference between consecutive terms is always 16. Therefore, the sequence can be described as an arithmetic sequence with a common difference of 16.

### Step 3: General form of an arithmetic sequence
The general formula for the \( n \)-th term of an arithmetic sequence can be written as:

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

where \( d \) is the common difference and \( a_1 \) is the first term.

### Step 4: Substitute the known values
In our case:
- \( a_1 = 15 \)
- \( d = 16 \)

Plugging these values into the formula gives:

\[
a_n = 15 + (n-1) \cdot 16
\]

### Step 5: Simplify the expression
Now, we simplify this expression:

\[
a_n = 15 + 16(n - 1)
\]
\[
= 15 + 16n - 16
\]
\[
= 16n - 1
\]

### Conclusion
Thus, the explicit formula for the sequence \( a_n \) is:

\[
\boxed{16n - 1}
\]