lesson 3: Arithmetic Sequences and Series
Introduction
Fill in the blanks to continue the pattern you see in this list: $2,\;7,\;6,\;11,\;10,\;15,\;\_\_\,,\;\_\_\,,\,\_\_\,,\;\_\_\,, \ldots $
In mathematics, a $sequence$ is a list of numbers, usually with an identifiable pattern or a connection to real data — for example, the list of expected temperatures for the next ten days, or a sequence listing the profits that a business earned year by year.
The numbers in the sequence are called $terms$ — for example, in the sequence above, 2 is the first term. Sometimes this pattern is very simple: 2,4,6,8…. Other times it might be harder to spot, like the problem above. Or, it might be an unusual type of pattern — 0,1,0,1,1,0,1,1,1,0,1,1,1,1….
Come up with the next two numbers in the sequence $1,3,6,10, \ldots $ , and a reason for why they should be the two numbers.
How about the next two numbers in the sequence $2,5,10,13,26,29, \ldots $ ?
Make up a sequence that you think will challenge your neighbor — then see if they can find the pattern.
A sequence starts with the numbers 5, 15, 35.
. Describe a pattern that this sequence might be following, and give the next three terms according to the pattern.
. Now describe a different pattern that the sequence might be following, and give the next three terms according to the new pattern.
The patterns you’ve seen so far have often been related to the differences between terms, such as the patterns in problems 1 and 2. When the pattern in a sequence is simply that the difference is always the same, the sequence is called an $arithmetic sequence$. For example, 7,9,11,13… is one of these.
Some of the sequences below are arithmetic, and some are not. For each sequence, only a few terms are given. Identify whether or not the sequence appears to be arithmetic. Also for each sequence find a pattern and fill in the missing terms.
. 3, 6, 12, 24, ___, ___, …
. 4, 11, 18, ___, ___, ….
. 9, ___, 17, ___, 25, ….
. 12, 13, 15, 16, 18, ___, ___,….
. 47, 43, 39, ___, 31, ___, ___, ….
. 13, 16, 22, 24, 28, 36, ___, ___, ___....
Development
In a sequence, it’s not only the numbers that matter—the order of the numbers is important too. For instance, sequence “A” — 20,10,40,30,60,50… — follows quite a different pattern from sequence “B” — 10,20,30,40,50,60…. — even though they contain all the same numbers.
In the first sequence, 30 is the 4th term, while in the second sequence, 30 is the 3rd term. To make this type of thing easier to write, we’ll introduce some notation.
In the first sequence, we write “${A_4} = 30$ ” (pronounced “A sub four” or simply “A four”) to mean that the sequence’s 4th term is 30. Then the first sequence is called $\{ {A_n}\} $ (because it is a set that consists of the terms ${A_1}$ , ${A_2}$ , ${A_3}$ , etc) and the second sequence is called $\{ {B_n}\} $ .
Fill in the blanks according to the sequences above.
. ${A_1}$ =_____.
. ${B_4}$ =_____.
. ${B_{\_\_\_\_}}$ = 30.
. ${A_{\_\_\_\_}}$ = 70.
One last piece of notation you’ll need to know is that, in an arithmetic sequence, the constant difference between terms is often called “d”. In the sequence below, $d = 2$ . 7, 9, 11, 13, . . .
In an arithmetic sequence called $\{ {T_n}\} $ , if ${T_1} = 13$ , and $d = 3$ , then what does ${T_{10}}$ equal? What does ${T_{100}}$ equal?
In an arithmetic sequence, ${T_4} = 8$ and ${T_5} = 11$ . Find the first three terms of the sequence.
An arithmetic sequence starts with ${T_1} = 4$ , ${T_2} = 9$ , … Which term will equal 99?
What might the next two terms be in the sequence 4, 6, 9…? Find several different possible patterns.
Make up five sequences all beginning with 1, 2, but with a different third term. Look at what others in your class came up with, as seeing different patterns can give you ideas for making up some really interesting sequences.
Consider the sequence 5, 10, 6, 12, 8, 16, 12, 24, ...
. Find the repeating pattern, and write the next 3 terms.
. It looks like all of the numbers after the “5” are even — will the sequence ever produce another odd number? Why or why not?
Consider the three sequences below. Sequences $\{ {D_n}\} $ and $\{ {E_n}\} $ below have some missing information, but you can assume that they are arithmetic.
|
n |
1 |
2 |
3 |
4 |
|
${A_n}$ |
12 |
19 |
26 |
33 |
|
n |
2 |
4 |
6 |
8 |
10 |
|
${D_n}$ |
? |
41 |
? |
48 |
? |
|
n |
9 |
11 |
13 |
15 |
17 |
|
${E_n}$ |
? |
6 |
? |
34 |
? |
. Which sequence, $\{ {D_n}\} $ or $\{ {E_n}\} $ , has the same value of d as sequence $\{ {A_n}\} $ ?
. Plot the information from the three sequences on a graph (put n on the horizontal axis). Put all three on the same graph. How does what you see fit with your answer to part a?
In your thermometer, the mercury reaches a height of 46 mm when the temperature is 70 degrees Fahrenheit, and a height of 48 mm when the temperature is 73 degrees.
. Think about these data as an arithmetic sequence where n is the temperature, and ${H_n}$ is the height at that temperature. Note that “…” is used here to indicate that this table is not showing you values of n and ${H_n}$ where n is less than 70 or greater than 75, but such values still exist.
ill in the table below:
|
n |
… |
70 |
71 |
72 |
73 |
74 |
75 |
… |
|
${H_n}$ |
… |
46 |
48 |
… |
. What is the value of d for this sequence?
. What would the height be if the temperature were at 106 degrees Fahrenheit?
. What would the height be if the temperature were at 65 degrees Fahrenheit?
In an arithmetic sequence, the constant difference is sometimes referred to as the rate of change of the sequence. Any idea why it might be so called?
If a sequence has ${T_5} = 40$ but it has a rate of change of -6, find ${T_4}$ , ${T_6}$ , and ${T_{20}}$ .
How would you find the value of d for the arithmetic sequence below? (Remember that “…” is used to indicate that there are other value(s) of n and ${T_n}$ that are not listed in this table but certainly exist!) Also write an equation for ${T_b}$ in terms of ${T_a}$ , a, b and d.
|
n |
… |
a |
… |
b |
… |
|
${T_n}$ |
… |
${T_a}$ |
… |
${T_b}$ |
… |
Practice
Another thermometer has a different relationship between height and temperature. For this thermometer, the equation ${H_n} = 0.3n + 17.1$ expresses the height ${H_n}$ (in mm) of the mercury in terms of the temperature n, in degrees Fahrenheit.
. What’s the height when the temperature is 50 degrees?
. If you think of ${H_n}$ as the nth term of a sequence, what’s the first term?
. What’s the value of d for the sequence $\{ {H_n}\} $ ?
. What would be the temperature if the height of the mercury were 20.7 mm ?
. What would be the temperature if the height of the mercury were 3 mm?
An arithmetic sequence $\{ {T_n}\} $ has $d = 5.5$ , and ${T_{12}}$ = 48.
. Find ${T_4}$ .
. For what value of n would ${T_n}$ = 130.5?
Each part below gives information about a sequence $\{ {A_n}\} $ . In each part, find the value of the unknown in the table for $\{ {B_n}\} $ so that the two sequences have the same rate of change.
(Remember that “…” is used to indicate that there are other value(s) of n , ${A_n}$ , and ${B_n}$ that are not listed in a table but certainly exist. Thus in the first table of part a., n can be less than 6, can be 7 or 8, and can be greater than 9, even though those values of n aren’t listed explicitly in the table.)
.
|
n |
… |
6 |
… |
9 |
… |
|
${A_n}$ |
… |
40 |
… |
82 |
… |
|
n |
… |
4 |
5 |
… |
|
${B_n}$ |
… |
a |
48 |
… |
.
|
n |
… |
16 |
… |
22 |
… |
|
${A_n}$ |
… |
50 |
… |
23 |
… |
|
n |
… |
1 |
… |
19 |
… |
|
${B_n}$ |
… |
b |
… |
40 |
… |
.
|
n |
… |
9 |
… |
31 |
… |
|
${A_n}$ |
… |
40.5 |
… |
71.3 |
… |
|
n |
… |
4 |
… |
14 |
… |
|
${B_n}$ |
… |
4 |
… |
c |
… |
.
|
n |
… |
10 |
… |
12 |
… |
|
${A_n}$ |
… |
12 |
… |
20 |
… |
|
n |
… |
30 |
… |
q |
… |
|
${B_n}$ |
… |
9 |
… |
29 |
… |