MM - Arithmetic & Geometric Sequences Lesson

Arithmetic & Geometric Sequences

Sequences and series arise in many classical mathematics problems as well as in more recently investigated mathematics, such as fractals. In this lesson, we are going to explore arithmetic sequences, both in recursive form and explicit form, as well as the sum of arithmetic series. Here are the formulas for the recursive and explicit forms of arithmetic sequences, as well as the recursive and explicit forms of geometric sequences.

Arithmetic & Geometric Sequences
Arithmetic Sequences	Geometric Sequences
Explicit Form	Explicit Form
$LaTeX: a_n=a_1+\left(n-1\right)\cdot d$ $a_n=a_1+\left(n-1\right)\cdot d$	$LaTeX: a_n=a_1\cdot r^{\left(n-1\right)}$ $a_n=a_1\cdot r^{\left(n-1\right)}$
Recursive Form	Recursive Form
$LaTeX: a_n=a_{n-1}+d$ $a_n=a_{n-1}+d$	$LaTeX: a_n=r\cdot a_{n-1}$ $a_n=r\cdot a_{n-1}$

A sequence could be considered a function whose domain is a set of consecutive integers. If the domain is not specified, it begins at 1. Finite sequences contain a last term, whereas infinite sequences continue without stopping. When the terms of a sequence are added, the resulting expression is a series.

Let's investigate some of the interesting patterns that arise when investigating and manipulating different figures.

Some of these shapes are called a fractal. Fractals are geometric patterns that are repeated at ever smaller increments. The fractal in this problem is a special type of snowflake, called the Koch snowflake. At each stage, the middle third of each side is replaced with an equilateral triangle. (See the diagram.)

If you were to take a large piece of paper, and construct an equilateral triangle with side lengths of 9 inches. On each side, you would locate the middle third. You would then construct a new equilateral triangle in that spot and erase the original part of the triangle that now forms the base of the new, smaller equilateral triangle.

Now consider each of the sides of the snowflake. How long is each side? Locate the middle third of each of these sides. How long would one-third of the side be? Construct new equilateral triangles at the middle of each of the sides.

How many sides are there to the snowflake now? Note that every side should be the same length. You would then continue the process a few more times.

Now the first three columns in the chart need to be completed.

	Number of Segments	Length of each Segment (in)	Perimeter (in)
Stage 1	3	9	27
Stage 2	12	3	36
Stage 3	48	1	48

Next, let's consider the number of segments in the successive stages.

The number of segments is 4 times the number in the previous stage, so this is a geometric sequence. In a geometric sequence, consecutive terms differ by a common ratio. In this case, that ratio is 4. If you make a plot of the segments, it will produce an exponential graph. So, the explicit formula will be the equation of an exponential function.

If we write a recursive and explicit formula for the number of segments at each stage, it will look like the following

equation image indicator $LaTeX: a_1=3 \\ a_n=4\cdot a_{n-1}\:\&\:a_n=\left(4^{n-1}\right)$ $a_1=3 \\ a_n=4\cdot a_{n-1}\:\&\:a_n=\left(4^{n-1}\right)$

The common ratio of the geometric sequence is 4.

We then use this equation to find the 7^th term, 12^th term, and 16^th term of the sequence.

equation image indicator $LaTeX: a_7=3\left(4^{7-1}\right)=12,288 \\ a_{12}=3\left(4^{12-1}\right)=12,582,912 \\ a_{16}=3\left(4^{16-1}\right)=3,221,225,472$ $a_7=3\left(4^{7-1}\right)=12,288 \\ a_{12}=3\left(4^{12-1}\right)=12,582,912 \\ a_{16}=3\left(4^{16-1}\right)=3,221,225,472$

Watch the following videos to investigate arithmetic sequences, which will allow you to better investigate the example below.

Video 1 explores three examples of arithmetic sequences and how to show them in both explicit form and recursive (recurring) form.

In the first 2:11 of video 2, a specific term is found by using the formula of an arithmetic sequence from a pattern of geometric shapes. From 2:12 to 8:44, the first example shows a derivation a sequence, domain, and range from a linear function. The second example shows the derivation of the domain and range from a sequence. The third example shows the derivation of the common difference, domain, and range from an arithmetic sequence. From 8:45 to the end, an example is shown that derives a recurrence relation (recursive formula) and an explicit formula from an arithmetic sequence in a table.

In the first 3:23 of video 3, an example shows the derivation of a recurrence relation (recursive formula) and an explicit formula from the graph of an arithmetic sequence. From 3:24 until the end, two examples are explored that find the next few terms in a geometric sequence, given the first 3 or 4 terms in that sequence. They define a geometric sequence as a recurrence relation and an explicit formula. They also make use of a TI-84 calculator to help in the process.

In the first 2:30 of video 4, the example shown creates a sequence from a function. From 2:31 to 4:53, a table is given and a recurrence relation (recursive formula) and explicit definition (formula) are derived. From 4:54 to 6:58, a graph is given and a recurrence relation (recursive formula) and explicit definition (formula) are derived. From 6:58 to 10:13, two examples are explored where a specific term in an arithmetic sequence is found. From 10:14 to the end, two examples are explored where a specific term in a geometric sequence is found.

In the first 4:21 of video 5, an application of a geometric sequence is used in a future value situation. From 4:22 to 6:46, an example of a geometric sequence is used in a situation involving how the number of people increases at a graduation party. From 6:47 to the end, an example of a geometric sequence being derived from a binary tree graph.

It's now time for us to practice Arithmetic and Geometric Sequences.

Arithmetic and Geometric Series

There are occasions where we need to find the sum of a finite arithmetic series and a finite geometric series , as well as the sum of an infinite geometric series. Here are the formulas for each of these series. Now, as you watch the following videos, you will explore arithmetic and geometric series, as well as the rationale behind them.

Arithmetic and Geometric Series
Arithmetic Series	Geometric Series
Summation Formula	Sum of Finite Series
$LaTeX: S_n=\frac{n}{2}\left(a_1+a_n\right)$ $S_n=\frac{n}{2}\left(a_1+a_n\right)$	$LaTeX: S_n=\frac{a_{1\left(1-r^n\right)}}{1-r}$ $S_n=\frac{a_{1\left(1-r^n\right)}}{1-r}$
	Sum of Infinite Series
	$LaTeX: S_n=\frac{a_1}{1-r},\:if\:-1<r<1$ $S_n=\frac{a_1}{1-r},\:if\:-1<r<1$

In Video 1, the first 3:00 introduces and derives the formula for the sum of an arithmetic series. From 3:01 to the end, two examples are shown that find the sum of arithmetic series.

In the first 10:15 of video 2, four examples on how to find a specific term in an arithmetic series, given the sum of the series. From 10:16 to 16:18, two examples are shown using the sum of an arithmetic series formula. From 16:19 to the end, there is an example exploring an application problem using the arithmetic series formula.

In Video 3, the first 4:39 introduces and derives the formula for the sum of a geometric series. From 4:40 to 6:39, an example is explored finding the sum of the first 14 terms of geometric series, when the first five terms are given. From 6:39 the end, an example is explored finding the sum of a set of terms in a geometric series, when the number of terms is not known.

In Video 4, the first 10:09 explores four examples using the sum of a finite geometric series formula. The first example (labeled number 3) finds the sum of the first 11 terms of a geometric series, given the first term and the common ratio. The second example (labeled number 4) finds the first term of a geometric series, given the sum of the first 11 terms and the common ratio. The third and fourth examples (labeled numbers 7 and 8) finds the sum of the first 5 and 11 terms respectively of a geometric series, given that the series is written in sigma notation. From 10:10 to 15:33, two examples (labeled examples 9 and 10) are shown finding the sum of an infinite geometric series. From 15:33 to the end, two examples (labeled examples 11 and 12) are shown finding the sum of an infinite geometric series, given the problems are written in sigma notation.

In Video 5, the first 8:05 explores four examples (labeled numbers 13 and 14 and another 13 and 14) that convert repeating decimal numbers into fractions using the sum of an infinite geometric series formula. From 8:06 to 9:27, explores one example (labeled number 15) that converts a "special" repeating decimal number into a whole number using the sum of an infinite geometric series formula. From 9:27 to the end, explores two examples (labeled examples 16 and 16) that are application problems using the sum of a finite geometric series formula.

It's now time for us to practice Arithmetic and Geometric Series.

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