# 11.2: Sequences and Their Notations

- Page ID
- 114113

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In this section, you will:

- Write the terms of a sequence defined by an explicit formula.
- Write the terms of a sequence defined by a recursive formula.
- Use factorial notation.

A video game company launches an exciting new advertising
campaign. They predict the number of online visits to their
website, or hits, will double each day. The model they are using
shows 2 hits the first day, 4 hits the second day, 8 hits the third
day, and so on. See __Table
1__.

Day |
1 | 2 | 3 | 4 | 5 | … |

Hits |
2 | 4 | 8 | 16 | 32 | … |

**Table** **1**

If their model continues, how many hits will there be at the end of the month? To answer this question, we’ll first need to know how to determine a list of numbers written in a specific order. In this section, we will explore these kinds of ordered lists.

### Writing the Terms of a Sequence Defined by an Explicit Formula

One way to describe an ordered list of numbers is as
a **sequence**. A sequence is a function whose
domain is a subset of the counting numbers. The sequence
established by the number of hits on the website is

{2,4,8,16,32,…}.{2,4,8,16,32,…}.

The ellipsis (…) indicates that the sequence continues
indefinitely. Each number in the sequence is called
a **term**. The first five terms of this sequence
are 2, 4, 8, 16, and 32.

Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.

One type of formula is an explicit formula, which defines the terms of a sequence using their position in the sequence. Explicit formulas are helpful if we want to find a specific term of a sequence without finding all of the previous terms. We can use the formula to find the nth term of the sequence, where nn is any positive number. In our example, each number in the sequence is double the previous number, so we can use powers of 2 to write a formula for the nthnth term.

The first term of the sequence is 21=2,21=2, the second term is 22=4,22=4, the third term is 23=8,23=8, and so on. The nthnth term of the sequence can be found by raising 2 to the nthnth power. An explicit formula for a sequence is named by a lower case letter a,b,c...a,b,c... with the subscript n.n. The explicit formula for this sequence is

an=2n.an=2n.

Now that we have a formula for the nthnth term of the
sequence, we can answer the question posed at the beginning of this
section. We were asked to find the number of hits at the end of the
month, which we will take to be 31 days. To find the number of hits
on the last day of the month, we need to find the
31^{st} term of the sequence. We will substitute 31
for nn in the formula.

a31=231 =2,147,483,648a31=231 =2,147,483,648

If the doubling trend continues, the company will get 2,147,483,6482,147,483,648 hits on the last day of the month. That is over 2.1 billion hits! The huge number is probably a little unrealistic because it does not take consumer interest and competition into account. It does, however, give the company a starting point from which to consider business decisions.

Another way to represent the sequence is by using a table. The
first five terms of the sequence and the nthnth term of
the sequence are shown in __Table
2__.

nn |
1 | 2 | 3 | 4 | 5 | nn |

nthnth term of the
sequence, anan |
2 | 4 | 8 | 16 | 32 | 2n2n |

**Table** **2**

Graphing provides a visual representation of the sequence as a
set of distinct points. We can see from the graph
in __Figure
1__ that the number of hits is rising at an exponential
rate. This particular sequence forms an exponential function.

Figure **1**

Lastly, we can write this particular sequence as

{2,4,8,16,32,…,2n,…}.{2,4,8,16,32,…,2n,…}.

A sequence that continues indefinitely is called
an **infinite sequence**. The domain of an
infinite sequence is the set of counting numbers. If we consider
only the first 10 terms of the sequence, we could write

{2,4,8,16,32,…,2n,…,1024}.{2,4,8,16,32,…,2n,…,1024}.

This sequence is called a **finite
sequence** because it does not continue
indefinitely.

A **sequence** is a function whose domain
is the set of positive integers. A **finite
sequence** is a sequence whose domain consists of only
the first nn positive integers. The numbers in a sequence
are called **terms**. The
variable aa with a number subscript is used to represent
the terms in a sequence and to indicate the position of the term in
the sequence.

a1,a2,a3,…,an,…a1,a2,a3,…,an,…

We call a1a1 the first term of the
sequence, a2a2 the second term of the
sequence, a3a3 the third term of the sequence, and so on.
The term anan is called the **nth term of the
sequence**, or the general term of the sequence.
An **explicit formula** defines
the nthnth term of a sequence using the position of the
term. A sequence that continues indefinitely is
an **infinite sequence**.

**Does a sequence always have to begin
with** a1?a1?

*No. In certain problems, it may be
useful to define the initial term as a0a0 instead
of a1.a1. In these problems, the domain of the function
includes 0.*

**Given an explicit formula, write the
first** nn **terms of a
sequence.**

- Substitute each value of nn into the formula. Begin with n=1n=1 to find the first term, a1.a1.
- To find the second term, a2,a2, use n=2.n=2.
- Continue in the same manner until you have identified all nn terms.

**EXAMPLE 1**

#### Writing the Terms of a Sequence Defined by an Explicit Formula

Write the first five terms of the sequence defined by the explicit formula an=−3n+8.an=−3n+8.

**Answer**-

#### Analysis

The sequence values can be listed in a table. A table, such
as __Table
3__, is a convenient way to input the function into a
graphing utility.

nn |
1 | 2 | 3 | 4 | 5 |

anan |
5 | 2 | –1 | –4 | –7 |

**Table** **3**

A graph can be made from this table of values. From the graph
in __Figure
2__, we can see that this sequence represents a linear
function, but notice the graph is not continuous because the domain
is over the positive integers only.

Figure **2**

Write the first five terms of the sequence defined by the explicit formula tn=5n−4.tn=5n−4.

#### Investigating Alternating Sequences

Sometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to finding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show increase or decrease as nn increases. Let’s take a look at the following sequence.

{2,−4,6,−8}{2,−4,6,−8}

Notice the first term is greater than the second term, the second term is less than the third term, and the third term is greater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret the sequence.

**Given an explicit formula with alternating terms, write
the first nn terms of a sequence.**

- Substitute each value of nn into the formula. Begin with n=1n=1 to find the first term, a1.a1. The sign of the term is given by the (−1)n(−1)n in the explicit formula.
- To find the second term, a2,a2, use n=2.n=2.
- Continue in the same manner until you have identified all nn terms.

**EXAMPLE 2**

#### Writing the Terms of an Alternating Sequence Defined by an Explicit Formula

Write the first five terms of the sequence.

an=(−1)nn2n+1an=(−1)nn2n+1

**Answer**-

#### Analysis

The graph of this function, shown in __Figure
3__, looks different from the ones we have seen previously in
this section because the terms of the sequence alternate between
positive and negative values.

Figure **3**

**In Example
2, does the (–1) to the power
of** nn

**account for the oscillations of signs?**

*Yes, the power might
be* n,n+1,n−1,n,n+1,n−1, *and so on, but any odd powers will result in a negative
term, and any even power will result in a positive term.*

Write the first five terms of the sequence.

an=4n(−2)nan=4n(−2)n

#### Investigating Piecewise Explicit Formulas

We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions. Recall that a piecewise function is a function defined by multiple subsections. A different formula might represent each individual subsection.

**Given an explicit formula for a piecewise function,
write the first nn terms of a sequence**

- Identify the formula to which n=1n=1 applies.
- To find the first term, a1,a1, use n=1n=1 in the appropriate formula.
- Identify the formula to which n=2n=2 applies.
- To find the second term, a2,a2, use n=2n=2 in the appropriate formula.
- Continue in the same manner until you have identified all nn terms.

**EXAMPLE 3**

#### Writing the Terms of a Sequence Defined by a Piecewise Explicit Formula

Write the first six terms of the sequence.

an={n2n3if nis not divisible by 3if nis divisible by 3an={ n2if nis not divisible by 3n3if nis divisible by 3

**Answer**-

#### Analysis

Every third point on the graph shown in __Figure
4__ stands out from the two nearby points. This occurs
because the sequence was defined by a piecewise function.

Figure **4**

Write the first six terms of the sequence.

an={2n35n2if nis oddif nis evenan={ 2n3if nis odd5n2if nis even

#### Finding an Explicit Formula

Thus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the nthnth term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases.

**Given the first few terms of a sequence, find an
explicit formula for the sequence.**

- Look for a pattern among the terms.
- If the terms are fractions, look for a separate pattern among the numerators and denominators.
- Look for a pattern among the signs of the terms.
- Write a formula for anan in terms of n.n. Test your formula for n=1,n=2,n=1,n=2, and n=3.n=3.

**EXAMPLE 4**

#### Writing an Explicit Formula for
the *n*th Term of a
Sequence

Write an explicit formula for the nthnth term of each sequence.

- ⓐ {−211,313,−415,517,−619,…}{ −211,313,−415,517,−619,… }
- ⓑ {−225,−2125,−2625,−23,125,−215,625,…}{−225,−2125,−2625,−23,125,−215,625,…}
- ⓒ {e4,e5,e6,e7,e8,…}{e4,e5,e6,e7,e8,…}

**Answer**-

Write an explicit formula for the nthnth term of the sequence.

{9,−81,729,−6,561,59,049,…}{9,−81,729,−6,561,59,049,…}

Write an explicit formula for the nthnth term of the sequence.

{−34,−98,−2712,−8116,−24320,...}{ −34,−98,−2712,−8116,−24320,... }

Write an explicit formula for the nthnth term of the sequence.

{1e2, 1e, 1, e, e2,...}{ 1e2, 1e, 1, e, e2,... }

### Writing the Terms of a Sequence Defined by a Recursive Formula

Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.

Each term of the Fibonacci sequence depends on the terms that
come before it. The Fibonacci sequence cannot easily be written
using an explicit formula. Instead, we describe the sequence using
a **recursive formula**, a formula that defines
the terms of a sequence using previous terms.

A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining anan in terms of preceding terms. For example, suppose we know the following:

a1=3an=2an−1−1 for n≥2a1=3an=2an−1−1 for n≥2

We can find the subsequent terms of the sequence using the first term.

a1=3a2=2a1−1=2(3)−1=5a3=2a2−1=2(5)−1=9a4=2a3−1=2(9)−1=17a1=3a2=2a1−1=2(3)−1=5a3=2a2−1=2(5)−1=9a4=2a3−1=2(9)−1=17

So the first four terms of the sequence are {3,5,9,17}{ 3,5,9,17 } .

The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.

a1=1a2=1an=an−1+an−2 for n≥3a1=1a2=1an=an−1+an−2 for n≥3

To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so

a10=a9+a8=34+21=55a10=a9+a8=34+21=55

A **recursive formula** is a formula that
defines each term of a sequence using preceding term(s). Recursive
formulas must always state the initial term, or terms, of the
sequence.

**Must the first two terms always be given in a recursive
formula?**

*No. The Fibonacci sequence defines
each term using the two preceding terms, but many recursive
formulas define each term using only one preceding term. These
sequences need only the first term to be defined.*

**Given a recursive formula with only the first term
provided, write the first nn terms of a
sequence.**

- Identify the initial term, a1,a1, which is given as part of the formula. This is the first term.
- To find the second term, a2,a2, substitute the initial term into the formula for an−1.an−1. Solve.
- To find the third term, a3,a3, substitute the second term into the formula. Solve.
- Repeat until you have solved for the nthnth term.

**EXAMPLE 5**

#### Writing the Terms of a Sequence Defined by a Recursive Formula

Write the first five terms of the sequence defined by the recursive formula.

a1=9an=3an−1−20, for n≥2a1=9an=3an−1−20, for n≥2

**Answer**-

Write the first five terms of the sequence defined by the recursive formula.

a1=2an=2an−1+1, for n≥2a1=2an=2an−1+1, for n≥2

**Given a recursive formula with two initial terms, write
the first nn terms of a sequence.**

- Identify the initial term, a1,a1, which is given as part of the formula.
- Identify the second term, a2,a2, which is given as part of the formula.
- To find the third term, substitute the initial term and the second term into the formula. Evaluate.
- Repeat until you have evaluated the nthnth term.

**EXAMPLE 6**

#### Writing the Terms of a Sequence Defined by a Recursive Formula

Write the first six terms of the sequence defined by the recursive formula.

a1=1a2=2an=3an−1+4an−2, for n≥3a1=1a2=2an=3an−1+4an−2, for n≥3

**Answer**-

Write the first 8 terms of the sequence defined by the recursive formula.

a1=0a2=1a3=1an=an−1an−2+an−3, for n≥4a1=0a2=1a3=1an=an−1an−2+an−3, for n≥4

### Using Factorial Notation

The formulas for some sequences include products of consecutive
positive integers. **nn factorial**, written
as n!,n!, is the product of the positive integers from 1
to n.n. For example,

4!=4⋅3⋅2⋅1=245!=5⋅4⋅3⋅2⋅1=1204!=4⋅3⋅2⋅1=245!=5⋅4⋅3⋅2⋅1=120

An example of formula containing a factorial is an=(n+1)!.an=(n+1)!. The sixth term of the sequence can be found by substituting 6 for n.n.

a6=(6+1)!=7!=7⋅6⋅5⋅4⋅3⋅2⋅1=5040a6=(6+1)!=7!=7·6·5·4·3·2·1=5040

The factorial of any whole number nn is n(n−1)!n(n−1)! We can therefore also think of 5!5! as 5⋅4!.5⋅4!.

** n factorial** is a mathematical
operation that can be defined using a recursive formula. The
factorial of n,n, denoted n!,n!, is defined for
a positive integer nn as:

0!=11!=1n!=n(n−1)(n−2)⋯(2)(1), for n≥20!=11!=1n!=n(n−1)(n−2)⋯(2)(1), for n≥2

The special case 0!0! is defined as 0!=1.0!=1.

**Can factorials always be found using a
calculator?**

*No. Factorials get large very
quickly—faster than even exponential functions! When the output
gets too large for the calculator, it will not be able to calculate
the factorial.*

**EXAMPLE 7**

#### Writing the Terms of a Sequence Using Factorials

Write the first five terms of the sequence defined by the explicit formula an=5n(n+2)!.an=5n(n+2)!.

**Answer**-

#### Analysis

__
Figure 7__ shows the graph of the sequence. Notice that,
since factorials grow very quickly, the presence of the factorial
term in the denominator results in the denominator becoming much
larger than the numerator as nn increases. This means the
quotient gets smaller and, as the plot of the terms shows, the
terms are decreasing and nearing zero.

Figure **7**

Write the first five terms of the sequence defined by the explicit formula an=(n+1)!2n.an=(n+1)!2n.

Access this online resource for additional instruction and practice with sequences.

### 11.1 Section Exercises

#### Verbal

__
1__.

Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?

2.

Describe three ways that a sequence can be defined.

__
3__.

Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

4.

What happens to the terms anan of a sequence when there is a negative factor in the formula that is raised to a power that includes n?n? What is the term used to describe this phenomenon?

__
5__.

What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

#### Algebraic

For the following exercises, write the first four terms of the sequence.

6.

an=2n−2an=2n−2

__
7__.

an=−16n+1an=−16n+1

8.

an=−(−5)n−1an=−(−5)n−1

__
9__.

an=2nn3an=2nn3

10.

an=2n+1n3an=2n+1n3

__
11__.

an=1.25⋅(−4)n−1an=1.25⋅(−4)n−1

12.

an=−4⋅(−6)n−1an=−4⋅(−6)n−1

__
13__.

an=n22n+1an=n22n+1

14.

an=(−10)n+1an=(−10)n+1

__
15__.

an=−(4⋅(−5)n−15)an=−(4⋅(−5)n−15)

For the following exercises, write the first eight terms of the piecewise sequence.

16.

an={(−2)n−2(3)n−1if nis evenif nis oddan={ (−2)n−2if nis even(3)n−1if nis odd

__
17__.

an={n22n+1n2−5if n≤5if n>5an={ n22n+1if n≤5n2−5if n>5

18.

an={(2n+1)22nif nis divisible by 4if nis not divisible by 4an={ (2n+1)2if nis divisible by 42nif nis not divisible by 4

__
19__.

an={−0.6⋅5n−12.5⋅(−2)n−1if nis prime or 1if nis compositean={ −0.6⋅5n−1if nis prime or 12.5⋅(−2)n−1if nis composite

20.

an={4(n2−2)n2−24if n≤3or n> 6if 3<n≤6an={ 4(n2−2)if n≤3or n> 6n2−24if 3<n≤6

For the following exercises, write an explicit formula for each sequence.

__
21__.

4, 7, 12, 19, 28,…4, 7, 12, 19, 28,…

22.

−4,2,−10,14,−34,…−4,2,−10,14,−34,…

__
23__.

1,1,43,2,165,…1,1,43,2,165,…

24.

0,1−e11+e2,1−e21+e3,1−e31+e4,1−e41+e5,…0,1−e11+e2,1−e21+e3,1−e31+e4,1−e41+e5,…

__
25__.

1,−12,14,−18,116,…1,−12,14,−18,116,…

For the following exercises, write the first five terms of the sequence.

26.

a1=9,an=an−1+na1=9,an=an−1+n

__
27__.

a1=3,an=(−3)an−1a1=3,an=(−3)an−1

28.

a1=−4,an=an−1+2nan−1−1a1=−4,an=an−1+2nan−1−1

__
29__.

a1=−1,an=(−3)n−1an−1−2a1=−1,an=(−3)n−1an−1−2

30.

a1=−30,an=(2+an−1)(12)na1=−30,an=(2+an−1)(12)n

For the following exercises, write the first eight terms of the sequence.

__
31__.

a1=124,a2=1,an=(2an−2)(3an−1)a1=124,a2=1,an=(2an−2)(3an−1)

32.

a1=−1,a2=5,an=an−2(3−an−1)a1=−1,a2=5,an=an−2(3−an−1)

__
33__.

a1=2,a2=10,an=2(an−1+2)an−2a1=2,a2=10,an=2(an−1+2)an−2

For the following exercises, write a recursive formula for each sequence.

34.

−2.5,−5,−10,−20,−40,…−2.5,−5,−10,−20,−40,…

__
35__.

−8,−6,−3,1,6,…−8,−6,−3,1,6,…

36.

2,4,12,48,240,…2,4,12,48,240,…

__
37__.

35,38,41,44,47,…35,38,41,44,47,…

38.

15,3,35,325,3125,⋯15,3,35,325,3125,⋯

For the following exercises, evaluate the factorial.

__
39__.

6!6!

40.

(126)!(126)!

__
41__.

12!6!12!6!

42.

100!99!100!99!

For the following exercises, write the first four terms of the sequence.

__
43__.

an=n!n2an=n!n2

44.

an=3⋅n!4⋅n!an=3⋅n!4⋅n!

__
45__.

an=n!n2−n−1an=n!n2−n−1

46.

an=100⋅nn(n−1)!an=100⋅nn(n−1)!

#### Graphical

For the following exercises, graph the first five terms of the indicated sequence

__
47__.

an=(−1)nn+nan=(−1)nn+n

48.

an={4+n2n3+nif nis evenif nis oddan={ 4+n2nif nis even3+nif nis odd

__
49__.

a1=2,an=(−an−1+1)2a1=2,an=(−an−1+1)2

50.

a1=1,an=an−1+8a1=1,an=an−1+8

__
51__.

an=(n+1)!(n−1)!an=(n+1)!(n−1)!

For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.

52.

__
53__.

54.

For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

__
55__.

56.

#### Technology

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

- On the home screen, key in the value for the initial
term a1a1 and press
**[ENTER]**. - Enter the recursive formula by keying in all numerical values
given in the formula, along with the key strokes
**[2ND] ANS**for the previous term an−1.an−1. Press**[ENTER]**. - Continue pressing
**[ENTER]**to calculate the values for each successive term.

For the following exercises, use the steps above to find the indicated term or terms for the sequence.

__
57__.

Find the first five terms of the
sequence a1=87111a1=87111, an=43an−1+1237.an=43an−1+1237. Use
the >**Frac** feature to give fractional
results.

58.

Find the 15^{th} term of the
sequence a1=625a1=625, an=0.8an−1+18.an=0.8an−1+18.

__
59__.

Find the first five terms of the sequence a1=2a1=2, an=2[(an−1)−1]+1.an=2[(an−1)−1]+1.

60.

Find the first ten terms of the sequence a1=8a1=8, an=(an−1+1)!an−1!.an=(an−1+1)!an−1!.

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61__.

Find the tenth term of the sequence a1=2a1=2, an=nan−1an=nan−1

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

- In the home screen, press
**[2ND] LIST**. - Scroll over to
**OPS**and choose**“seq(”**from the dropdown list. Press**[ENTER]**. - In the line headed
**“Expr:”**type in the explicit formula, using the [X,T,θ,n][X,T,θ,n] button for nn - In the line headed
**“Variable:”**type in the variable used on the previous step. - In the line headed
**“start:”**key in the value of nn that begins the sequence. - In the line headed
**“end:”**key in the value of nn that ends the sequence. - Press
**[ENTER]**3 times to return to the home screen. You will see the sequence syntax on the screen. Press**[ENTER]**to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

Using a TI-83, do the following.

- In the home screen, press
**[2ND] LIST**. - Scroll over to
**OPS**and choose**“seq(”**from the dropdown list. Press**[ENTER]**. - Enter the items in the
order
**“Expr”**,**“Variable”**,**“start”**,**“end”**separated by commas. See the instructions above for the description of each item. - Press
**[ENTER]**to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

62.

List the first five terms of the sequence an=−289n+53.an=−289n+53.

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63__.

List the first six terms of the sequence an=n3−3.5n2+ 4.1n−1.52.4n.an=n3−3.5n2+ 4.1n−1.52.4n.

64.

List the first five terms of the sequence an=15n⋅(−2)n−147an=15n⋅(−2)n−147

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65__.

List the first four terms of the sequence an=5.7n+0.275(n−1)!an=5.7n+0.275(n−1)!

66.

List the first six terms of the sequence an=n!n.an=n!n.

#### Extensions

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67__.

Consider the sequence defined by an=−6−8n.an=−6−8n. Is an=−421an=−421 a term in the sequence? Verify the result.

68.

What term in the sequence an=n2+4n+42(n+2)an=n2+4n+42(n+2) has the value 41?41? Verify the result.

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69__.

Find a recursive formula for the sequence 1, 0, −1, −1, 0, 1, 1,
0, −1, −1, 0, 1, 1, .... (*Hint*: find
a pattern for anan based on the first two terms.)

70.

Calculate the first eight terms of the sequences an=(n+2)!(n−1)!an=(n+2)!(n−1)! and bn=n3+3n2+2n,bn=n3+3n2+2n, and then make a conjecture about the relationship between these two sequences.

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71__.

Prove the conjecture made in the preceding exercise.