Question

1. suppose you have the sequence 2, 5, 8, 11, ...
2. how could you write this as a function?
3. what does slope mean in this situation?

Explanation:

Step1: Identify the type of sequence

The sequence 2, 5, 8, 11, ... is an arithmetic - sequence with a first term \(a_1 = 2\) and a common difference \(d=3\). The general formula for an arithmetic sequence is \(a_n=a_1+(n - 1)d\). We can rewrite it as a function \(f(n)\) where \(n\) represents the position of the term in the sequence.

Step2: Write the function

Substitute \(a_1 = 2\) and \(d = 3\) into the formula. We get \(f(n)=2+(n - 1)\times3\). Simplify the expression: \(f(n)=2 + 3n-3=3n - 1\), where \(n\in\mathbb{N}\) (natural numbers, \(n = 1,2,3,\cdots\)).

Step3: Interpret the slope

If we consider the function \(y = f(n)=3n - 1\) as a linear function (where \(n\) is the independent variable and \(y\) is the dependent variable), in the slope - intercept form \(y=mx + b\) (\(m\) is the slope and \(b\) is the y - intercept), here \(m = 3\). In the context of the arithmetic sequence, the slope represents the common difference between consecutive terms. That is, for each increase of 1 in the position \(n\) of the term in the sequence, the value of the term increases by 3.

Answer:

1. \(f(n)=3n - 1\), \(n\in\mathbb{N}\)
2. The slope represents the common difference between consecutive terms of the arithmetic sequence, which is 3. So for each increase of 1 in the position of the term in the sequence, the value of the term increases by 3.