Question

the explict formula below describes a linear function where n is a positive integer
f(n)=5n-1
what is the first term and the recursive formula that describes the same function

Answer 1

The explicit formula given is \( f(n) = 5n - 1 \). To convert this to a recursive formula, we first need to determine the first term of the function.

1. **Finding the First Term:**
For \( n = 1 \):
\[
f(1) = 5(1) - 1 = 5 - 1 = 4
\]
Thus, the first term \( f(1) \) is 4.

2. **Finding the Recursive Formula:**
The recursive formula for a linear function can generally be described as:
\[
f(n) = f(n-1) + d
\]
where \( d \) is the common difference.

From the explicit formula, we can see that the linear function increases by 5 for each increment of \( n \) (since the coefficient of \( n \) is 5). Therefore, the common difference \( d = 5 \).

So, the recursive formula will be:
- Base case: \( f(1) = 4 \)
- Recursive case: \( f(n) = f(n-1) + 5 \) for \( n > 1 \)

Putting this together, the first term is \( 4 \), and the recursive formula can be expressed as:

\[
\begin{align*}
f(1) & = 4 \\
f(n) & = f(n-1) + 5 \quad \text{for } n > 1
\end{align*}
\]