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# Sequences

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Arithmetic and Geometric Sequences
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A sequence is a list of numbers with a common pattern.  The common pattern in an arithmetic sequence is that the same number is added or subtracted to each number to produce the next number.  The common pattern in a geometric sequence is that the same number is multiplied or divided to each number to produce the next number.

Are all sequences arithmetic or geometric?

#### Guidance

A sequence is just a list of numbers separated by commas.  A sequence can be finite or infinite.  If the sequence is infinite, the first few terms are followed by an ellipsis $(\ldots)$ indicating that the pattern continues forever.

An infinite sequence : $1, 2, 3, 4, 5, \ldots$

A finite sequence: $2, 4, 6, 8$

In general, you describe a sequence with subscripts that are used to index the terms.  The  $k^{th}$ term in the sequence is $a_k$ .

$a_1, a_2, a_3, a_4,\ldots, a_k, \ldots$

Arithmetic sequences are defined by an initial value  $a_1$ and a common difference $d$ .

$a_1 &=a_1 \\a_2 &=a_1+d \\a_3 &=a_1+2d \\a_4 &=a_1+3d \\& \ \ \vdots \\a_n &=a_1+(n-1)d$

Geometric sequences are defined by an initial value  $a_1$ and a common ratio $r$ .

$a_1 &=a_1\\a_2 &=a_1 \cdot r\\a_3 &=a_1\cdot r^2\\a_4 &=a_1 \cdot r^3\\& \ \ \vdots \\a_n &=a_1 \cdot r^{n-1}$

If a sequence does not have a common ratio or a common difference, it is neither an arithmetic or a geometric sequence.  You should still try to figure out the pattern and come up with a formula that describes it.

Example A

For each of the following three sequences, determine if it is arithmetic, geometric or neither.

1. $0.135, 0.189, 0.243, 0.297, \ldots$
2. $\frac{2}{9}, \frac{1}{6}, \frac{1}{8}, \ldots$
3. $0.54, 1.08, 3.24, \ldots$

Solution:

1. The sequence is arithmetic because the common difference is 0.054.
2. The sequence is geometric because the common ratio is $\frac{3}{4}$ .
3. The sequence is not arithmetic because the differences between consecutive terms are 0.54 and 2.16 which are not common.  The sequence is not geometric because the ratios between consecutive terms are 2 and 3 which are not common.

Example B

For the following sequence, determine the common ratio or difference, the next three terms, and the  $2013^{th}$ term.

$\frac{2}{3}, \frac{5}{3}, \frac{8}{3}, \frac{11}{3}, \ldots$

Solution:  The sequence is arithmetic because the difference is exactly 1 between consecutive terms.  The next three terms are $\frac{14}{3}, \frac{17}{3}, \frac{20}{3}$ .  An equation for this sequence would is:

$a_n=\frac{2}{3}+(n-1) \cdot 1$

Therefore, the $2013^{th}$ term requires 2012 times the common difference added to the first term.

$a_{2013}=\frac{2}{3}+2012 \cdot 1=\frac{2}{3}+\frac{6036}{3}=\frac{6038}{3}$

Example C

For the following sequence, determine the common ratio or difference and the next three terms.

$\frac{2}{3}, \frac{4}{9}, \frac{6}{27}, \frac{8}{81}, \frac{10}{243}, \ldots$

Solution:  This sequence is neither arithmetic nor geometric.  The differences between the first few terms are $-\frac{2}{9}, -\frac{2}{9}, -\frac{10}{81}, -\frac{14}{243}$ .  While there was a common difference at first, this difference did not hold through the sequence.  Always check the sequence in multiple places to make sure that the common difference holds up throughout.

The sequence is also not geometric because the ratios between the first few terms are $\frac{2}{3}, \frac{1}{2}, \frac{4}{9}$ .  These ratios are not common.

Even though you cannot get a common ratio or a common difference, it is still possible to produce the next three terms in the sequence by noticing the numerator is an arithmetic sequence with starting term of 2 and a common difference of 2.  The denominators are a geometric sequence with an initial term of 3 and a common ratio of 3.  The next three terms are:

$\frac{12}{3^6}, \frac{14}{3^7}, \frac{16}{3^8}$

Concept Problem Revisited

Example C shows that some patterns that use elements from both arithmetic and geometric series are neither arithmetic nor geometric.  Two famous sequences that are neither arithmetic nor geometric are the Fibonacci sequence and the sequence of prime numbers.

Fibonacci Sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$

Prime Numbers:  $2, 3, 5, 7, 11, 13, 17, 19, 23, \ldots$

#### Vocabulary

A sequence is a list of numbers separated by commas.

The common pattern in an arithmetic sequence is that the same number is added or subtracted to each number to produce the next number.  This is called the common difference.

The common pattern in a geometric sequence is that the same number is multiplied or divided to each number to produce the next number.  This is called the common ratio .

#### Guided Practice

1. What is the tenth term in the following sequence?

$-12, 6, -3, \frac{3}{2}, \ldots$

2. What is the tenth term in the following sequence?

$-1, \frac{2}{3}, \frac{7}{3}, 4, \frac{17}{3}, \ldots$

3. Find an equation that defines the  $a_k$ term for the following sequence.

$0, 3, 8, 15, 24, 35, \ldots$

1. The sequence is geometric and the common ratio is $-\frac{1}{2}$ .  The equation is $a_n=-12 \cdot \left(-\frac{1}{2}\right)^{n-1}$ .  The tenth term is:

$-12 \cdot \left(-\frac{1}{2}\right)^9=\frac{3}{128}$

2. The pattern might not be immediately recognizable, but try ignoring the  $\frac{1}{3}$ in each number to see the pattern a different way.

$-3, 2, 7, 12, 17, \ldots$

You should see the common difference of 5.  This means the common difference from the original sequence is $\frac{5}{3}$ .  The equation is $a_n=-1+(n-1)\left(\frac{5}{3}\right)$ .  The  $10^{th}$ term is:

$-1+9 \cdot \left(\frac{5}{3}\right)=-1+3 \cdot 5=-1+15=14$

3. The sequence is not arithmetic nor geometric.  It will help to find the pattern by examining the common differences and then the common differences of the common differences.  This numerical process is connected to ideas in calculus.

0, 3, 8, 15, 24, 35

3, 5, 7, 9, 11

2, 2, 2, 2

Notice when you examine the common difference of the common differences the pattern becomes increasingly clear.  Since it took two layers to find a constant function, this pattern is quadratic and very similar to the perfect squares.

$1, 4, 9, 16, 25, 36, \ldots$

The  $a_k$ term can be described as $a_k=k^2-1$

#### Practice

Use the sequence  $1, 5, 9, 13, \ldots$ for questions 1-3.

1. Find the next three terms in the sequence.

2. Find an equation that defines the  $a_k$ term of the sequence.

3. Find the  $150^{th}$ term of the sequence.

Use the sequence  $12, 4, \frac{4}{3}, \frac{4}{9}, \ldots$ for questions 4-6.

4. Find the next three terms in the sequence.

5. Find an equation that defines the  $a_k$ term of the sequence.

6. Find the  $17^{th}$ term of the sequence.

Use the sequence $10, -2, \frac{2}{5}, -\frac{2}{25}, \ldots$ for questions 7-9.

7. Find the next three terms in the sequence.

8. Find an equation that defines the  $a_k$ term of the sequence.

9. Find the  $12^{th}$ term of the sequence.

Use the sequence  $\frac{7}{2}, \frac{9}{2}, \frac{11}{2}, \frac{13}{2}, \ldots$ for questions 10-12.

10. Find the next three terms in the sequence.

11. Find an equation that defines the  $a_k$ term of the sequence.

12. Find the $314^{th}$ term of the sequence.

13. Find an equation that defines the $a_k$ term for the sequence  $4, 11, 30, 67, \ldots$

14. Explain the connections between arithmetic sequences and linear functions.

15. Explain the connections between geometric sequences and exponential functions.