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Introduction
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Types of Arrangements
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Two More Methods
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Some Applications

Section 3.1 - Sequences

In English, a sequence means an ordered process or set of steps. The mathematical definition is almost identical: an ordered list of numbers, usually based on some kind of formula.

Sequence

A sequence is a list of numbers that we write {an}. Each number in the sequence is called a term. The n represents the number of the term. For example, a3 is the third term and a6 is the sixth term.

Example 1 - Finding the Terms of a Sequence

List the first four terms of the sequence whose nth term is an = 4n + 3n.

To find the terms of the sequence, we just replace the n's with the number of the term. For example, the first term would be

$$a_n=4n+3^n$$ $$a_1 = 4 \cdot 1 + 3^1$$ $$a_1 = 4 + 3 = 7$$

To find the second term, I'll replace n with 2, to find the third term, I'll replace n with 3, etc.

$$a_2 = 4 \cdot 2 + 3^2 = 8 + 9 = 17$$ $$a_3 = 4 \cdot 3 + 3^3 = 12 + 27 = 39$$ $$a_4 = 4 \cdot 4 + 3^4 = 16 + 81 = 97$$

So the first four terms in the sequence are 7, 17, 39, 97.

Example 2 - Finding the Terms of a Sequence

Find the 11th term of the sequence whose nth term is $a_n = (-1)^n (n + 2)$.

To find the 11th term, I'll replace each of the n's in the formula with 11.

$$a_n = (-1)^n (n + 2)$$ $$a_{11} = (-1)^{11} (11 + 2)$$ $$a_{11} = -1 \cdot (13)$$ $$a_{11} = -13$$

Arithmetic Sequences

Arithmetic Sequences

An arithmetic sequence is a sequence where each term is equal to the previous term plus a constant value d. If a1 is the first term of the sequence then the nth term is an = a1 + d(n - 1). The constant d is called the interval or the common difference.

The key to identifying an arithmetic sequence is looking at the difference between each term. For example, look at

2, 6, 10, 14, 18, 22, 26

Notice how each number is 4 more than the one before it:

6 = 2 + 4
10 = 6 + 4
14 = 10 + 4
. . .

The number that gets added at each step is the d in the definition. The first term in this sequence is 2, i.e. a1 = 2, so the formula for this sequence is $$a_n = a_1+d(n-1)$$ $$a_n=2+4(n-1)$$

Technically, we should also say that n has to between 1 and 7 because the sequence only has seven terms.

Example 3 - Arithmetic Sequences

Show that the sequence 6, 11, 16, 21, 26 is an arithmetic sequence.

The easiest way to do this is to subtract each term from the one after it and confirm that you get the same number every time. That number is the sequence's interval.

11 - 6 = 5
16 - 11 = 5
21 - 16 = 5
26 - 21 = 5

Because we got the same result all four times, the sequence is an arithmetic sequence with d = 5.

Example 4 - Arithmetic Sequences

Find the 12th term in the arithmetic sequence whose first term is -5 and whose interval is 3.5.

The nth term of the sequence is

an = -5 + 3.5(n - 1)

The 12th term is

a12 = -5 + 3.5(12 - 1)
a12 = 33.5

Example 5 - Arithmetic Sequences

Find a formula for the nth term of the sequence 7, 10, 13, 16, 19, 22.

The first step in answering this question is recognizing that this is an arithmetic sequence. You can see that each term in the sequence is 3 more than the one before it. That makes this an arithmetic sequence with d = 3 and a1 = 7. If we substitute those values into the arithmetic sequence formula we get

an = 7 + 3(n - 1)

for values of n between 1 and 6.

We'll sometimes write the value of n using an inequality:

an = 7 + 3(n - 1), 1 ≤ n ≤ 6

Geometric Sequences

Geometric Sequences

A geometric sequence is a sequence whose first term is a1 and whose nth term is an = a1rn - 1. r is called the sequence's common ratio.

Geometric sequences are very similar to arithmetic sequences except each term gets multiplied by a number instead of having something added to it. For example, look at

2, 6, 18, 54, 162

In this case, each number is three times the one before it.

6 = 2 · 3
18 = 6 · 3
54 = 18 · 3
. . .

Example 6 - Geometric Sequences

Show that the sequence 2, 4, 8, 16, 32, 64 is a geometric sequence.

The procedure here is very similar to the one we used with an arithmetic sequence. First, I'll divide each term by the term before it.

4 ÷ 2 = 2
8 ÷ 4 = 2
16 ÷ 8 = 2
32 ÷ 16 = 2
64 ÷ 32 = 2

I got the same number each time which confirms that this is a geometric sequence with 2 as its common ratio.

Example 7 - Geometric Sequences

Find the 10th term of the geometric sequence whose first term is 3 and whose common ratio is -4.

If I substitute r = -4 and r1 = 3 into the formula for a geometric sequence, I get that the nth term is

$$a_n = 3 \cdot (-4)^{n-1}$$

So the 10th term must be

$$a_{10} = 3 \cdot (-4)^{10-1}$$ $$a_{10} = 3 \cdot (-4)^9$$ $$a_{10} = 3 \cdot (-524288)$$ $$a_{10} = -1572864$$

Example 8 - Geometric Sequences

Find a formula for the nth term of the geometric sequence 3, 3/2, 3/4, 3/8.

The first step in answering this question is recognizing that this is a geometric sequence. It's a little tricky with this one because the common ratio is a fraction but you can find it if you divide the second term by the first.

$$\frac{\frac{3}{2}}{3} = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2}$$

If we substitute those 1/2 for r and 3 for a1 into the geometric sequence formula, we get

$$a_n = 3\cdot \left(\frac{1}{2}\right)^{n-1}, 1 \le n \le 4$$

Exercises

Exercises

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a
1. 1, 5, 9, 13, 17, 21, 25
2. 10, 20, 40, 70, 110, 160
3. 1, 3, 9, 27, 81, 243
4. 5, 1, -3, -7, -11, -15, -19, -23, -27
5. 3, 10, 17, 24, 30, 37, 44, 51
6. 6, -12, 24, -48, 96
7. 16, 8, 4, 2, 1
8. 17, 16, 15, 14, 13, 12, 11, 10
9. $a_n = 3n + 4$
10. $a_n = 2^{n} + 1$
11. $a_n = 1 / (n + 1)$
12. $a_n = (n - 1)!$
13. $a_n = 4 \cdot 3^{n}$
14. $a_n = n! / (4n)$
15. $a_n = n^{2} + 3n - 4$
16. $a_n = 2n^{2}$
17. $a_n = n! + (n - 1)!$
18. a geometric sequence with r = -3 and a1 = 2.
19. a geometric sequence with r = 1/3 and a1 = 4.
20. a geometric sequence with r = 2 and a2 = 8.
21. a geometric sequence with a2 = 7 and a3 = 14.
22. an arithmetic sequence with d = 12 and a1 = 0.
23. an arithmetic sequence with d = -2 and a3 = 4.
24. an arithmetic sequence with a4 = 5 and a5 = 7.5.
25. an arithmetic sequence with d = 2 and a1 = 17.
26. 2, 6, 18, 54, 162, 486, 1458
27. 3.5, 4.7, 5.9, 7.1, 8.3, 9.5
28. 12, 7, 2, -3, -8
29. 18, 5.4, 1.62, 0.486, 0.1458
30. 12, 13, 14, 15, 16, 17, 18, 19, 20
31. 12, 4, 4/3, 4/9
32. 3, -6, 12, -24, 48, -96
33. 18, 15, 12, 9, 6, 3, 0

Analysis

1. Suppose you have two arithmetic sequences with the same number of terms and you add them together. Show that the new sequence is also an arithmetic sequence. What is its interval?
2. The sum of two geometric sequences is not a geometric sequence. Show this is true by giving a counterexample, i.e. take two geometric sequences, add their terms together and show that the sequence you get isn't a geometric sequence.
3. What's the 15th term of the arithmetic sequence that starts with 10, 12.5, 15, 17.5.
4. Suppose that you get 1 penny on the first of March. Every day after that, you get twice the number of pennies that you got on the day before. How much do you get on March 31?
5. Write the first six terms of the arithmetic sequence with $a_1=3$ and $d=2$ and the geometric sequence with $b_1 = 3$ and $r=2$. Compare the results, e.g. which type of sequence grows faster?

Technical Writing

1. Explain the difference between a geometric and an arithmetic sequence.
2. Describe the process for identifying an arithmetic sequence.
3. Why does the formula for a geometric sequence have n - 1 in it instead of just n? (Think about how a1 has to equal the first term of the sequence.)