Discrete Math - The MKD Remix

A linear function is one that has a constant rate of change.

\begin{array}{|l|c|c|c|c|c|} & x & 1 & 2 & 3 & 4 & 5 \\ \hline \\ & y & 1 & 3 & 5 & 7 & 9 \end{array}

The table above displays a function with independent variable \(x\) and dependent variable \(y\).

Notice that the value of \(y\) increases by \(2\) for each increase in \(x\) by \(1\). The rate of change of this function is \(2\); this corresponds to the slope \(m\) of the continuous line that passes through the points with \(xy\)-coordinates given in the table.

The vertical intercept \(b\) (in this case, the y-_intercept ) is the \(y\)-value that corresponds to \(x = 0\), that is, \((0,\,b)\) is on the same continuous line as the points represented in the table. In this example, \(0\) is not a value of \(x\), but we can still find the vertical intercept by _subtracting \(1\) from the smallest \(x\)-value and subtracting \(2\) from the corresponding \(y\)-value , which tells us that the point \((0,\,-1)\) lies on the same continuous line as the points represented in the table.

The equation of the linear function determined by the points in the table is \(y = 2 \cdot x + (-1)\), which can be written more simply as \(y = 2x - 1\). This also is the equation of the continuous line that passes through the points with \(xy\)-coordinates given in the table, but the linear function can be restricted to a smaller domain as needed by the context where it is being used, for example, we may only need to use inputs \(x\) from the set of positive integers or possibly just the set \(\{ 1,\,2,\,3,\,4,\,5 \}\).

An arithmetic sequence or arithmetic progression is a sequence of numbers \(a_{0}, \, a_{1}, \, a_{2}, \, \ldots\) such that there is a constant \(d\) so that \[ a_{i+1}-a_{i} = d \text{ for all } i \in \mathbb{N}\] The constant \(d\) is called the common difference of the sequence. The sequence can be infinite, defined for every \(i \in \mathbb{N},\) or finite, defined only for \(i \in \mathbb{N}\) less than some greatest index \(n\).

As an example, the sequence \(1, 4, 7, 10, 13, 16\) is a finite arithmetic sequence with common difference 3 and index set \(\{ 0, 1, 2, 3, 4, 5 \}.\) We can extend that sequence to an infinite arithmetic sequence \(1, 4, 7, 10, 13, 16, \ldots\) using a recursive definition \[a_{0} = 1, \text{ and } a_{i+1} = a_{i} + 3 \text{ for integer } \in \mathbb{N} \]

Notice that there is also a nonrecursive definition for this sequence: Since the difference between two consecutive terms of the sequence is always \(d\) the points \((i, \, a_{i})\) must lie on a line in the xy -plane. The slope of this line is \(d\) and the y -intercept of the line is the initial value \(a_{0},\) so the arithmetic sequence can also be described as \[ a_{i}= d \cdot i + a_{0} \text{ for all } i \in \mathbb{N}\]

For the example \(1, 4, 7, 10, 13, 16, \ldots\), the nonrecursive definition is \[a_{i} = 3i + 1 \text{ for all } i \in \mathbb{N}\]

A geometric sequence or geometric progression is a sequence of numbers \(a_{0}, \, a_{1}, \, a_{2}, \, \ldots\) such that there is a constant \(r\) so that \[ a_{i+1} = r \cdot a_{i} \text{ for all } i \in \mathbb{N}\] The constant \(r\) is called the common ratio of the sequence. The sequence can be infinite, defined for every \(i \in \mathbb{N},\) or finite, defined only for \(i \in \mathbb{N}\) less than some greatest index \(n\).

As an example, the sequence \(5, 10, 20, 40, 80\) is a finite geometric sequence with common ratio 2 and index set \(\{ 0, 1, 2, 3, 4 \}.\) We can extend that sequence to an infinite geometric sequence \(5, 10, 20, 40, 80, \ldots\) using a recursive definition \[a_{0} = 5, \text{ and } a_{i+1} = a_{i} \cdot 2 \text{ for integer } i \in \mathbb{N} \]

There is also a nonrecursive definition for this sequence: Since the ratio between a term and its predecessor in the sequence is always \(r\) the points \((i, \, a_{i})\) must lie on the graph of an exponential function in the xy -plane. The y -intercept of the graph is the initial value \(a_{0},\) so the geometric sequence can be described as \[ a_{i} = a_{0} \cdot r^{i} \text{ for all } i \in \mathbb{N}\]

For the example \(5, 10, 20, 40, 80, \ldots\), the nonrecursive definition is \[a_{i} = 5 \cdot 2^{i} \text{ for all } i \in \mathbb{N}\]

MORE TO COME!