Question

finally, what is the general term definition for this sequence? 3, 5, 7, 9, ... recursive definition: \

$$\begin{cases} f(0) = 3 \\\\ f(n) = f(n - 1) + 2 \\end{cases}$$

general term definition: f(n) = ?n + \\ \\

Explanation:

Step1: Identify sequence type

The sequence \(3, 5, 7, 9, \dots\) is arithmetic with first term \(a_0 = 3\) and common difference \(d = 2\).

Step2: Recall arithmetic sequence formula

The general term of an arithmetic sequence is \(f(n)=f(0)+d\cdot n\), where \(f(0)\) is the initial term and \(d\) is the common difference.

Step3: Substitute values

Here, \(f(0) = 3\) and \(d = 2\), so \(f(n)=3 + 2n\), which can be written as \(f(n)=2n + 3\).

Answer:

The coefficient of \(n\) is \(2\) and the constant term is \(3\), so \(f(n)=\boldsymbol{2}n + \boldsymbol{3}\).