MFG Sequences

Section 9.1 Sequences

Subsection Definitions and Notation

Consider the following function: Gwynn would like to compete in a triathlon, but she needs to improve her swimming. She begins a training schedule in which she swims 20 laps a day for the first week and increases that number by 6 laps each week. The table below gives the first few values of the function.

Week	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$n$
Number of Laps	$20$	$26$	$32$	$38$	$44$	$50$	$56$	$62$	$f(n)$

The function makes sense only for input values that are positive integers. We would not ask how many laps Gwynn swims in week 4.63, or in week $-6\text{.}$

A function whose inputs are a set of successive positive integers is called a sequence. Most people think of a sequence as a list of objects in which the order is important, and we often present a mathematical sequence in just that way. The information in the table above can be displayed more simply by listing only the output values, in order:

\begin{equation*} 20,~~26,~~32,~~38,~~44,~~50,~~56,~~62 \end{equation*}

When we list them in this way, the output values are called the terms of the sequence. The input values are indicated implicitly by the position of the term. For example, the third term of the sequence, 32, is the value of $f(3)\text{.}$

Note 9.1.

We often use the notation $a_n$ instead of $f(n)$ to refer to the terms of a sequence. Thus, $a_1 = f(1),~~a_2=f(2),$ and so on.

Example 9.2.

Let $a_n$ be the number of seats in the $n^{th}$ row of a theater. Write an equation using subscript notation for each of the following statements.

The first row has 30 seats.
The twentieth row has 68 seats.
Row 18 has twice as many seats as row 2.
Row $n$ has $2n+28$ seats.
Row $n+1$ has 2 more seats than the previous row.

Solution.

$a_1$ represents the number of seats in row 1, so $a_1=30\text{.}$
$a_{20}$ represents the number of seats in row 20, so $a_{20}=68\text{.}$
$\displaystyle a_{18}=2a_2$
$\displaystyle a_n = 2n+28$
$a_{n+1} = 2+a_n\text{.}$ (Row $n$ is the row before row $n+1\text{.}$)

In Example 1, $a_n=2n+28$ is a formula in terms of $n$ for the number of seats in the $n^{th}$ row of the theater. The expression $2n+28$ is called the general term for the sequence.

Checkpoint 9.3. Practice 1.

Let $A_n$ be the number of dollars in a bank account at the end of its $n^{th}$year. Write an equation using subscript notation for each of the following statements.

The account had $1149.90 at the end of the second year.
At the end of year 4 the account had $85.75 more than at the end of year 3.
At the end of year 5 the account had 1.07 times as much as at the end of year 4.

Solution.

$\displaystyle A_2=1149.90$
$\displaystyle A_4=85.75+A_3$
$\displaystyle A_5=1.07A_4$

Example 9.4.

Find the first four terms in each sequence with the given general term.

$\displaystyle a_n=\dfrac{n(n+1)}{2}$
$\displaystyle a_n=(-1)^n 2^n$

Solution.

We evaluate each general term for successive values of $n\text{.}$

- $\displaystyle a_1=\dfrac{1(1+1)}{2}=1$
- $\displaystyle a_2=\dfrac{2(2+1)}{2}=3$
- $\displaystyle a_3=\dfrac{3(3+1)}{2}=6$
- $\displaystyle a_4=\dfrac{4(4+1)}{2}=10$
The first four terms are 1, 3, 6, and 10.
- $\displaystyle a_1=(-1)^1 2^1=-2$
- $\displaystyle a_2=(-1)^2 2^2=4$
- $\displaystyle a_3=(-1)^3 2^3=-8$
- $\displaystyle a_4=(-1)^4 2^4=16$
The first four terms are $-2, $ $4, $ $-8, $ and $16\text{.}$

Checkpoint 9.5. QuickCheck 1.

How is a sequence different from an ordinary function?

Its outputs are integers.
Its inputs are integers.
There is a formula for its outputs.
The outputs must be positive..

Checkpoint 9.6. Practice 2.

Find the first four terms in the sequence with the general term $b_n=(n+1)^2-n^2\text{.}$

Solution.

3, 5, 7, 9

Subsection Applications of Sequences

Sequences are useful for describing situations that have discrete values, rather than continuous values. For some savings accounts, interest is posted only once a year.

Example 9.7.

You deposit $8000 in a savings account that pays 5% interest compounded annually. How much money will be in the account at the end of each of the next 4 years?

Solution.

During the first year the account will earn 5% of $8000, or $0.05 (8000) = 400$ dollars. Thus, at the end of the first year the account will contain the original $8000 plus the $400 interest for a total of $8400.

At the end of the second year the account will have the $8400 from the previous year, plus 5% of $8400, or $0.05(8400) = 420$ dollars in interest. We can write this sum as

\begin{equation*} 8400 + 8400(0.05) = 8400 (1 + 0.05) = 8400(1.05)~~ \text{dollars.} \end{equation*}

Thus, at the end of the second year there will be $$8400(1.05) = $8820$ in the account.

In fact, each new balance is found by multiplying the previous balance by 1.05. The balances at the ends of the third and fourth years are

\begin{equation*} \$8820(1.05) = \$9261~~~~\text{and}~~~~\$9261(1.05) = \$9724.05 \end{equation*}

The annual balances form the sequence 8400, 8820, 9261, and 9724.05, as shown below.

$n$	$a_n$
1	8400
2	8820
3	9261
4	9724.05

Checkpoint 9.8. Practice 3.

You just finished your last cup of coffee and have 100 mg of caffeine in your system. For each hour that passes, the amount of caffeine in your system decreases by 14%. How much caffeine is in your system at the start of each of the next 4 hours?

Solution.

100 mg, 86 mg, 74 mg, 63.6 mg approximately

Checkpoint 9.9. QuickCheck 2.

The terms of a sequence

describe its formula.
are its boundary values.
are its outputs.
are either increasing or decreasing.

Example 9.10.

You deposit $8000 in a savings account that pays 5% simple annual interest. (The interest is earned only on the initial $8000 and not compounded.) How much money will be in the account at the end of each of the next 4 years?

Solution.

The account earns 5% of $8000 or $0.05(8000) = 400$ dollars during the first year. At the end of the first year the account will contain the original $8000 plus $400 interest for a total of $8400.

During the second year interest will be earned only on the initial $8000 deposited, for another 400 dollars. So at the end of the second year the account will have the $8400 from the previous year plus $400 interest, for a total of

\begin{equation*} 8400 + 400 = 8800 ~\text{dollars} \end{equation*}

In fact, the balance at the end of each succeeding year is found by adding $400 simple interest to the previous balance. The annual balances form the sequence 8400, 8800, 9200, and 9600, as shown below.

$n$	$b_n$
$1$	$8400$
$2$	$8800$
$3$	$9200$
$4$	$9600$

Checkpoint 9.11. Practice 4.

The financing agreement on your $12,000 car requires you to pay $1200 now and $208.87 a month for 5 years. What is the total amount you have paid after each of the first 4 payments?

Solution.

$1200, $1408.87, $1617.74, $1826.61

Example 9.12.

A regular polygon is a geometric figure in which all the sides are equal in length. For example, an equilateral triangle and a square are regular polygons. All the interior angles in a regular polygon are equal also. Find the general term $a_n$ for the sequence that gives the size of an interior angle in a regular polygon of $n$ sides.

$regular polygons$

Solution.

This sequence starts with $a_3$ because we cannot have a polygon with fewer than three sides. We already know the values of $a_3$ and $a_4\text{:}$ each angle in an equilateral triangle is $60\degree\text{,}$ and each angle in a square is $90\degree\text{.}$ We would like to find a formula for the size of the angles in any regular polygon.

If we can find the sum of all the angles in a regular polygon, we can divide by $n$ to find the size of each. (You can check that this idea works for the equilateral triangle and the square.) To find the sum of the angles, notice that any polygon can be partitioned into triangles, as shown below.

$regular polygons$

By sketching some examples, convince yourself that every polygon of $n$ sides (for $n \ge 3$) can be partitioned into $n-2$ triangles. Because the angles in every triangle add up to $180\degree\text{,}$ the sum of the angles in an $n$-sided polygon is $(n-2)$ times $180\degree\text{.}$ To find the size of just one of the angles, we divide the sum by $n\text{.}$ This gives us the general term of the sequence:

\begin{equation*} a_n = \dfrac{(n-2)180}{n} \end{equation*}

Subsection Recursively Defined Sequences

A sequence is defined recursively if each term of the sequence is defined in terms of its predecessors. For example, the sequence defined by

\begin{equation*} a_1=2,~~~~a_{n+1} = 3a_n - 2 \end{equation*}

is a recursive sequence. Its first four terms are

\begin{equation*} \begin{aligned}[t] a_1 \amp = 2\\ a_2 \amp = 3a_1 - 2 = 3(2)-2 = 4\\ a_3 \amp = 3a_2 - 2 = 3(4)-2 = 10\\ a_4 \amp = 3a_3 - 2 = 3(10)-2 = 28\\ \end{aligned} \end{equation*}

Example 9.13.

Make a table showing the first five terms of the recursive sequence

\begin{equation*} a_1 = -1,~~~~~~a_{n+1} = (a_n)^2-4 \end{equation*}

Solution.

The first term is given. We find each new term by using the recursive formula:

\begin{equation*} \begin{aligned}[t] a_2 \amp = (a_1)^2-4 = (-1)^2 - 4 = -3\\ a_3 \amp = (a_2)^2-4 = (-3)^2 - 4 = 5\\ a_4 \amp = (a_3)^2-4 = 5^2 - 4 = 21\\ a_5 \amp = (a_4)^2-4 = {21}^2-4 = 437\\ \end{aligned} \end{equation*}

The first five terms are 1, 3, 5, 21, and 437, as shown in the table.

$n$	$1$	$2$	$3$	$4$	$5$
$a_n$	$-1$	$-3$	$5$	$21$	$437$

The definition of a recursive sequence must include a starting point, usually the first term of the sequence, and a formula for calculating the next term of the sequence in terms of the previous term (or terms).

Example 9.14.

You deposit $8000 in a savings account that pays 5% interest compounded annually. Find a recursive definition for the sequence of account balances if the money is kept in the account for $n$ years.

Solution.

If we use the letter $a$ for that sequence, we have $a_1 = 8400\text{.}$ Each successive term was obtained by multiplying the previous term by 1.05, which means that

\begin{equation*} a_n = 1.05 a_{n-1} \end{equation*}

The sequence is determined by the recursive definition

\begin{equation*} a_1 = 8400,~~~~a_n = 1.05 a_{n-1} \end{equation*}

Checkpoint 9.15. Practice 5.

Find a recursive definition for the sequence in CheckPoint 9.8 above if you do not drink any more coffee for $n$ hours.

Solution.

$C_1 = 100\text{,}$ $C_{n+1} = 0.86 C_n$

Example 9.16.

You deposit $8000 in a savings account that pays 5% simple annual interest. Find a recursive definition for the sequence of account balances if the money is kept in the account for $n$ years.

Solution.

If we use the letter $b$ for that sequence, we have $b_1 = 8400\text{.}$ Each successive term was obtained by adding 400 to the previous term, which means that

\begin{equation*} b_n = b_{n-1} + 400 \end{equation*}

The sequence is determined by the recursive definition

\begin{equation*} b_1 = 8400,~~~b_n = b_{n-1} + 400 \end{equation*}

Checkpoint 9.17. QuickCheck 3.

If each output of a sequence is defined in terms of earlier outputs, we say the sequence is

general.
discrete.
ordered.
recursive.

Example 9.18.

Karen joins a savings plan in which she deposits $200 per month and receives 12% annual interest compounded monthly.

Find a recursively defined sequence that gives the amount of money in Karen’s account $n$ months later.
Find the first four terms of the sequence.

Solution.

In the first month Karen deposits $200, so $a_1 = 200\text{.}$ Each month thereafter Karen receives 1% interest (one-twelfth of 12% annual interest) on the previous month’s balance, and then adds $200 to the total. For example, before she makes her deposit in the second month the account has
\begin{equation*} 200 + 0.01 (200) = 1.01(200)~\text{dollars} \end{equation*}
She then adds $200 to this amount for a total of
\begin{equation*} a_2 = 1.01(200) + 200 ~\text{dollars} \end{equation*}
In general, after the $n^{th}$ deposit Karen’s account contains $a_n$ dollars. In the next month she earns 1% interest on that balance, giving her
\begin{equation*} a_n + 1.01 a_n = 1.01 a_n ~\text{dollars} \end{equation*}
Then she deposits another $200 for a total of
\begin{equation*} a_{n+1} = 1.01 a_n + 200 ~\text{dollars} \end{equation*}
Thus, the recursive sequence is defined by
\begin{equation*} a_1 = 200, ~~~ a_{n+1} = 1.01 a_n + 200 ~\text{dollars} \end{equation*}
We evaluate the general formula found in part (a) for $n = 1, 2, 3, 4:$
\begin{equation*} \begin{aligned}[t] a_1 \amp = 200\\ a_2 \amp = 1.01 a_1 + 200\\ \amp = 1.01 (200) + 200 = 402\\ a_3 \amp = 1.01 a_2 + 200\\ \amp = 1.01 (402) + 200 = 606.02\\ a_4 \amp = 1.01 a_3 + 200\\ \amp = 1.01 (606.02) + 200 = 812.08\\ \end{aligned} \end{equation*}

Subsection Section Summary

Subsubsection Vocabulary

Sequence
General term
Recursive sequence

Subsubsection CONCEPTS

A function whose inputs are a set of successive positive integers is called a sequence.
The output values are called the terms of the sequence.
A formula in terms of $n$ for the $n^{th}$ term of a sequence is called the general term of the sequence.
A sequence is defined recursively if each term of the sequence is defined in terms of its predecessors.

Subsubsection STUDY QUESTIONS

What distinguishes a sequence from an ordinary function?
What are the range values of a sequence called?
Give an example of a situation in which a sequence is more appropriate than a function whose domain is an interval of real numbers.
What is a recursively defined sequence? How is such a sequence defined?

Subsubsection SKILLS

Practice each skill in the Homework Problems listed.

Evaluate the general term #1–14, #21–26,
Use subscript notation #15–20
Evaluate a recursively defined sequence #27–34, #45–52
Write a formula for a sequence #35–44

Exercises Homework 9.1

Exercise Group.

For Problems 1–14, find the first four terms in the sequence whose general term is given.

1.

$a_n = n-5$

2.

$b_n = 2n-3$

3.

$c_n = \dfrac{n^2-2}{2}$

4.

$d_n = \dfrac{3}{n^2+1}$

5.

$s_n = 1+\dfrac{1}{n}$

6.

$t_n = \dfrac{n}{2n-1}$

7.

$u_n = \dfrac{n(n-1)}{2}$

8.

$v_n =\dfrac{5}{n(n+1)}$

9.

$w_n = (-1)^n$

10.

$A_n = (-1)^{n+1}$

11.

$B_n = \dfrac{(-1)^n (n-2)}{n}$

12.

$C_n = (-1)^{n-1} 3^{n+1}$

13.

$D_n = 1$

14.

$E_n = -1$

Exercise Group.

For Problems 15–20, suppose that $a_1, a_2, a_3,...$ is a sequence. Write an equation using subscript notation for each sentence.

15.

The first term of the sequence is $\dfrac{4}{3}\text{.}$

16.

The third term of the sequence is 8 more than the second term.

17.

The $n^{\text{th}}$ term is 3 times the previous term.

18.

The $n^{\text{th}}$ term is one third of the term that follows it.

19.

The $(n+1)^{\text{st}}$ term is one third of the term that follows it.

20.

The $(n+1)^{\text{st}}$ term is 3 times the previous term.

Exercise Group.

For Problems 21–26, find the indicated term for each sequence.

21.

$D_n = 2^n - n;~~~D_6$

22.

$E_n = \sqrt{n+1};~~~E_{11}$

23.

$x_n = \log n;~~~x_{26}$

24.

$y_n = \log (n+1);~~~y_9$

25.

$z_n = 2\sqrt{n};~~~z_{20}$

26.

$U_n = \dfrac{n+1}{n-1};~~~U_{17}$

Exercise Group.

For Problems 27-34, make a table showing the first five terms of the recursively defined sequence.

27.

$s_1=3;~~~s_n=s_{n-1}+2$

28.

$c_1=6;~~~c_n=c_{n-1}-4$

29.

$d_1=24;~~~d_{n+1}=\dfrac{-1}{2}d_n$

30.

$r_1=27;~~~r_{n+1}=\dfrac{2}{3}r_n$

31.

$t_1=1;~~~t_{n+1}=(n+1)t_n$

32.

$x_1=1;~~~x_{n+1}=(\dfrac{n}{n-1})x_n$

33.

$w_1 = 100;~~~w_n=1.10w_{n-1}+100$

34.

$q_1 = 100;~~~q_n=0.9q_{n-1}+100$

Exercise Group.

For Problems 35–42,

Make a table showing the first four terms of each sequence.
Write an equation to define the sequence recursively.

35.

A new car costs $14,000 and depreciates in value by 15% each year. How much is the car worth after $n$ years?

36.

Krishna takes a job as an executive secretary for $21,000 per year with a guaranteed 5% raise each year. What will his salary be after $n$ years?

37.

A long distance phone call costs $1.10 to make the connection and an additional $0.45 for each minute. What is the cost of a call that lasts $n$ minutes?

38.

Bettina earns $1000 per month plus $57 for each satellite dish that she sells. What is her monthly income when she sells $n$ satellite dishes?

39.

Geraldo inherits an annuity of $50,000 that earns 12% annual interest compounded monthly. If he withdraws $500 at the end of each month, what is the value of the annuity after $n$ months?

40.

Eve borrowed $18,000 for a new car at 6% annual interest compounded monthly. If she pays $400 per month toward the loan, how much does she owe after $n$ months?

41.

Majel must take 10 milliliters of a medication directly into her bloodstream at constant intervals. During each time interval her kidneys filter out 20% of the drug present just after the most recent dose. How much of the drug will be in her bloodstream after $n$ doses?

42.

A forest contains 64,000 trees. According to a new logging plan, each year 5% of the trees will be cut down and 16,000 new trees will be planted. How many trees will be in the forest after $n$ years?

43.

Draw three non-collinear points in the plane. (The points should not lie on the same line.) How many distinct lines are determined by the points? (In other words, how many different lines can you draw by choosing two of the points and joining them?)
Add a fourth point to your diagram. Now how many lines are determined?
Let $L_n$ stand for the number of distinct lines determined by non-collinear points. Make a table showing the first five terms of the sequence.
Find a recursive formula for the sequence $L_n\text{.}$

44.

Draw two distinct non-parallel lines in the plane. In how many points do the lines intersect?
Add a third line to your diagram that is not parallel to either of the first two lines. How many intersection points are there?
Let $P_n$ stand for the number of intersection points determined by lines in the plane, no two of which are parallel. Make a table showing the first five terms of the sequence.
Find a recursive formula for the sequence $P_n\text{.}$

45.

The Fibonacci sequence is found throughout nature. For example, the numbers of spirals in a sunflower or on a pineapple are elements of the Fibonacci sequence. It is named after the Italian mathematician Fibonnaci, who used it to model the growth of a population of rabbits. The Fibonacci sequence is defined recursively by

\begin{equation*} f_1=1,~~f_2=1,~~f_{n+2}=f_n + f_{n+1} \end{equation*}

Make a table showing the first 16 terms of the Fibonacci sequence.
Calculate the quotients $\dfrac{f_{n+1}}{f_n}$ for $n=1$ to $n=15\text{.}$ What do you observe? Now find a decimal approximation for the golden ratio, $\dfrac{1+\sqrt{5}}{2}\text{.}$

46.

The Lucas sequence is defined recursively by

\begin{equation*} L_1=2,~~L_2=1,~~L_{n+2}=L_n + L_{n+1} \end{equation*}

Find the first 10 terms of the Lucas sequence.
Calculate $(L_n)^2-L_n(L_{n+2})$ for $n=1$ to $n=8\text{.}$ What do you notice?

Exercise Group.

For Problems 47–52, use a calculator to evaluate a large number of terms for each recursive sequence. What happens to the terms as $n$ gets larger?

47.

$a_1=1;~~~a_n=\dfrac{1}{1+a_{n-1}}+1$

48.

$b_1=1;~~~b_n=\dfrac{2}{1+b_{n-1}}+1$

49.

$c_1=3;~~~c_n=\dfrac{\sqrt{1+c_{n-1}}}{2}$

50.

$d_1=8;~~~d_n=\dfrac{\sqrt{1+d_{n-1}}}{2}$

51.

$s_1=1;~~~s_n=\dfrac{1}{2}(s_{n-1} + \dfrac{4}{s_{n-1}})$

52.

$t_1=1;~~~t_n=\dfrac{1}{2}(t_{n-1} + \dfrac{9}{t_{n-1}})$

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Week	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(n\)
Number of Laps	\(20\)	\(26\)	\(32\)	\(38\)	\(44\)	\(50\)	\(56\)	\(62\)	\(f(n)\)

\(n\)	\(b_n\)
\(1\)	\(8400\)
\(2\)	\(8800\)
\(3\)	\(9200\)
\(4\)	\(9600\)

\(n\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(a_n\)	\(-1\)	\(-3\)	\(5\)	\(21\)	\(437\)