Precalculus

Ron Larson

Chapter 9

Sequences, Series, and Probability - all with Video Answers

Educators

Section 1

Sequences and Series

Problem 1

Fill in the blanks.
An ____ is a function whose domain is the set of positive integers.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 2

Fill in the blanks.
A sequence is a____ sequence when the domain of the function consists only of the first $n$ positive integers.

Aditya Sood

Numerade Educator

Problem 3

Fill in the blanks.
When you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, the sequence is defined ____.

Anurag Kumar

Numerade Educator

Problem 4

Fill in the blanks.
If $n$ is a positive integer, then $n$____ is defined as $n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdot \cdot(n-1) \cdot n$.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 5

Fill in the blanks.
For the sum $\sum_{i=1}^{n} a_{i}, i$ is the ___ of summation, $n$ is the ___limit of summation, and 1 is ___the limit of summation .

Anurag Kumar

Numerade Educator

Problem 6

Fill in the blanks.
The sum of the terms of a finite or infinite sequence is called a ____.

Jerry Zhang

Numerade Educator

Problem 7

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=4 n-7$$

AG

Ankit Gupta

Numerade Educator

Problem 8

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=-2 n+8$$

Anurag Kumar

Numerade Educator

Problem 9

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=(-1)^{n+1}+4$$

Anurag Kumar

Numerade Educator

Problem 10

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=1-(-1)^{n}$$

Anurag Kumar

Numerade Educator

Problem 11

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=(-2)^{n}$$

AG

Ankit Gupta

Numerade Educator

Problem 12

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=\left(\frac{1}{2}\right)^{n}$$

AG

Ankit Gupta

Numerade Educator

Problem 13

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=\frac{2}{3}$$

Anurag Kumar

Numerade Educator

Problem 14

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=6(-1)^{n+1}$$

Anurag Kumar

Numerade Educator

Problem 15

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=\frac{1}{3} n^{3}$$

Anurag Kumar

Numerade Educator

Problem 16

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=\frac{1}{n^{2}}$$

AG

Ankit Gupta

Numerade Educator

Problem 17

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=\frac{n}{n+2}$$

Anurag Kumar

Numerade Educator

Problem 18

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=\frac{6 n}{3 n^{2}-1}$$

Anurag Kumar

Numerade Educator

Problem 19

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=n(n-1)(n-2)$$

Anurag Kumar

Numerade Educator

Problem 20

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=n\left(n^{2}-6\right)$$

Anurag Kumar

Numerade Educator

Problem 21

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)$$

Anurag Kumar

Numerade Educator

Problem 22

Write the first five terms of the sequence. (Assume that $n \text { begins with } 1 .)$
$$a_{n}=\frac{(-1)^{n+1}}{n^{2}+1}$$

Anurag Kumar

Numerade Educator

Problem 23

Find the missing term of the sequence.
$$\begin{aligned}
&a_{n}=(-1)^{n}(3 n-2)\\
&a_{25}= \square
\end{aligned}$$

Anurag Kumar

Numerade Educator

Problem 24

Find the missing term of the sequence.
$$\begin{array}{l}
a_{n}=(-1)^{n-1}[n(n-1)] \\
a_{16}= \square
\end{array}$$

Anurag Kumar

Numerade Educator

Problem 25

Find the missing term of the sequence.
$$\begin{aligned}
&a_{n}=\frac{4 n}{2 n^{2}-3}\\
&a_{11}= \square
\end{aligned}$$

Anurag Kumar

Numerade Educator

Problem 26

Find the missing term of the sequence.
$$\begin{aligned}
&a_{n}=\frac{4 n^{2}-n+3}{n(n-1)(n+2)}\\
&a_{13}= \square
\end{aligned}$$

Anurag Kumar

Numerade Educator

Problem 27

Use a graphing utility to graph the first 10 terms of the sequence. (Assume that $n$ begins with 1 .)
$$a_{n}=\frac{2}{3} n$$

AG

Ankit Gupta

Numerade Educator

Problem 28

Use a graphing utility to graph the first 10 terms of the sequence. (Assume that $n$ begins with 1 .)
$$a_{n}=3 n+3(-1)^{n}$$

Anurag Kumar

Numerade Educator

Problem 29

Use a graphing utility to graph the first 10 terms of the sequence. (Assume that $n$ begins with 1 .)
$$a_{n}=16(-0.5)^{n-1}$$

AG

Ankit Gupta

Numerade Educator

Problem 30

Use a graphing utility to graph the first 10 terms of the sequence. (Assume that $n$ begins with 1 .)
$$a_{n}=8(0.75)^{n-1}$$

AG

Ankit Gupta

Numerade Educator

Problem 31

Use a graphing utility to graph the first 10 terms of the sequence. (Assume that $n$ begins with 1 .)
$$a_{n}=\frac{2 n}{n+1}$$

AG

Ankit Gupta

Numerade Educator

Problem 32

Use a graphing utility to graph the first 10 terms of the sequence. (Assume that $n$ begins with 1 .)
$$a_{n}=\frac{3 n^{2}}{n^{2}+1}$$

AG

Ankit Gupta

Numerade Educator

Problem 33

Match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).]
(GRAPHS CANNOT COPY)
$$a_{n}=\frac{8}{n+1}$$

Anurag Kumar

Numerade Educator

Problem 34

Match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).]
(GRAPHS CANNOT COPY)
$$a_{n}=\frac{8 n}{n+1}$$

Anurag Kumar

Numerade Educator

Problem 35

Match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).]
(GRAPHS CANNOT COPY)
$$a_{n}=4(0.5)^{n-1}$$

Anurag Kumar

Numerade Educator

Problem 36

Match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).]
(GRAPHS CANNOT COPY)
$$a_{n}=n\left(2-\frac{n}{10}\right)$$

Anurag Kumar

Numerade Educator

Problem 37

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$3,7,11,15,19, \dots$$

AG

Ankit Gupta

Numerade Educator

Problem 38

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$0,3,8,15,24, \dots$$

AG

Ankit Gupta

Numerade Educator

Problem 39

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$3,10,29,66,127, \dots$$

Anurag Kumar

Numerade Educator

Problem 40

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$91,82,73,64,55, \dots$$

Anurag Kumar

Numerade Educator

Problem 41

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$1,-1,1,-1,1, \ldots$$

AG

Ankit Gupta

Numerade Educator

Problem 42

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$1,3,1,3,1, . .$$

AG

Ankit Gupta

Numerade Educator

Problem 43

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$-\frac{2}{y}, \frac{3}{4},-\frac{4}{5}, \frac{5}{6},-\frac{6}{7}, \ldots$$

Anurag Kumar

Numerade Educator

Problem 44

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$\frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \dots$$

AG

Ankit Gupta

Numerade Educator

Problem 45

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$\frac{2}{1}, \frac{3}{3}, \frac{4}{5}, \frac{5}{7}, \frac{6}{9}, \dots$$

AG

Ankit Gupta

Numerade Educator

Problem 46

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$\frac{1}{3}, \frac{2}{9}, \frac{4}{27}, \frac{8}{81}, \dots$$

Anurag Kumar

Numerade Educator

Problem 47

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \dots$$

Anurag Kumar

Numerade Educator

Problem 48

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$2,3,7,25,121, \dots$$

Anurag Kumar

Numerade Educator

Problem 49

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$\frac{1}{1}, \frac{3}{1}, \frac{9}{2}, \frac{27}{6}, \frac{81}{24}, \ldots$$

Anurag Kumar

Numerade Educator

Problem 50

Write an expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. (Assume that $n \text { begins with } 1 .)$
$$\frac{2}{1}, \frac{6}{3}, \frac{24}{7}, \frac{120}{15}, \frac{720}{31}, \dots$$

Anurag Kumar

Numerade Educator

Problem 51

Write the first five terms of the sequence defined recursively.
$$a_{1}=28, \quad a_{k+1}=a_{k}-4$$

Anurag Kumar

Numerade Educator

Problem 52

Write the first five terms of the sequence defined recursively.
$$a_{1}=3, \quad a_{k+1}=2\left(a_{k}-1\right)$$

AG

Ankit Gupta

Numerade Educator

Problem 53

Write the first five terms of the sequence defined recursively.
$$a_{1}=81, \quad a_{k+1}=\frac{1}{3} a_{k}$$

Anurag Kumar

Numerade Educator

Problem 54

Write the first five terms of the sequence defined recursively.
$$a_{1}=14, \quad a_{k+1}=(-2) a_{k}$$

Anurag Kumar

Numerade Educator

Problem 55

Write the first five terms of the sequence defined recursively.
$$a_{0}=1, \quad a_{1}=2, \quad a_{k}=a_{k-2}+\frac{1}{2} a_{k-1}$$

Anurag Kumar

Numerade Educator

Problem 56

Write the first five terms of the sequence defined recursively.
$$a_{0}=-1, \quad a_{1}=1, \quad a_{k}=a_{k-2}+a_{k-1}$$

Anurag Kumar

Numerade Educator

Problem 57

Use the Fibonacci sequence. (See Example $5 .$ )
Write the first 12 terms of the Fibonacci sequence whose $n$ th term is $a_{n}$ and the first 10 terms of the sequence given by
$b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1$

Anurag Kumar

Numerade Educator

Problem 58

Use the Fibonacci sequence. (See Example $5 .$ )
Using the definition for $b_{n}$ in Exercise $57,$ show that $b_{n}$ can be defined recursively by
$b_{n}=1+\frac{1}{b_{n-1}}$

AG

Ankit Gupta

Numerade Educator

Problem 59

Write the first five terms of the sequence. (Assume that $n \text { begins with } 0 .)$
$$a_{n}=\frac{5}{n !}$$

Anurag Kumar

Numerade Educator

Problem 60

Write the first five terms of the sequence. (Assume that $n \text { begins with } 0 .)$
$$a_{n}=\frac{1}{(n+1) !}$$

Anurag Kumar

Numerade Educator

Problem 61

Write the first five terms of the sequence. (Assume that $n \text { begins with } 0 .)$
$$a_{n}=\frac{(-1)^{n}(n+3) !}{n !}$$

Anurag Kumar

Numerade Educator

Problem 62

Write the first five terms of the sequence. (Assume that $n \text { begins with } 0 .)$
$$a_{n}=\frac{(-1)^{2 n+1}}{(2 n+1) !}$$

Anurag Kumar

Numerade Educator

Problem 63

Simplify the factorial expression.
$$\begin{aligned}
&\frac{4 !}{6 !}\\
&(n+
\end{aligned}$$

Anurag Kumar

Numerade Educator

Problem 64

Simplify the factorial expression.
$$\frac{12 !}{4 ! \cdot 8 !}$$

AG

Ankit Gupta

Numerade Educator

Problem 65

Simplify the factorial expression.
$$\frac{(n+1) !}{n !}$$

Anurag Kumar

Numerade Educator

Problem 66

Simplify the factorial expression.
$$\frac{(2 n-1) !}{(2 n+1) !}$$

AG

Ankit Gupta

Numerade Educator

Problem 67

Find the sum.
$$\sum_{i=0}^{4} 3 i^{2}$$

Anurag Kumar

Numerade Educator

Problem 68

Find the sum.
$$\sum_{k=1}^{4} 10$$

AG

Ankit Gupta

Numerade Educator

Problem 69

Find the sum.
$$\sum_{i=3}^{5} \frac{1}{j^{2}-3}$$

Anurag Kumar

Numerade Educator

Problem 70

Find the sum.
$$\sum_{i=1}^{5}(2 i-1)$$

Anurag Kumar

Numerade Educator

Problem 71

Find the sum.
$$\sum_{k=2}^{5}(k+1)^{2}(k-3)$$

Anurag Kumar

Numerade Educator

Problem 72

Find the sum.
$$\sum_{i=1}^{4}\left[(i-1)^{2}+(i+1)^{3}\right]$$

Anurag Kumar

Numerade Educator

Problem 73

Find the sum.
$$\sum_{i=1}^{4} \frac{i !}{2^{i}}$$

Anurag Kumar

Numerade Educator

Problem 74

Find the sum.
$$\sum_{j=0}^{5} \frac{(-1)^{j}}{j !}$$

Anurag Kumar

Numerade Educator

Problem 75

Use a graphing utility to find the sum.
$$\sum_{k=0}^{4} \frac{(-1)^{k}}{k !}$$

AG

Ankit Gupta

Numerade Educator

Problem 76

Use a graphing utility to find the sum.
$$\sum_{k=0}^{4} \frac{(-1)^{k}}{k+1}$$

AG

Ankit Gupta

Numerade Educator

Problem 77

Use a graphing utility to find the sum.
$$\sum_{n=0}^{25} \frac{1}{4^{n}}$$

Anurag Kumar

Numerade Educator

Problem 78

Use a graphing utility to find the sum.
$$\sum_{n=0}^{10} \frac{n !}{2^{n}}$$

Anurag Kumar

Numerade Educator

Problem 79

Use sigma notation to write the sum.
$$\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\dots+\frac{1}{3(9)}$$

Anurag Kumar

Numerade Educator

Problem 80

Use sigma notation to write the sum.
$$\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\dots+\frac{5}{1+15}$$

Anurag Kumar

Numerade Educator

Problem 81

Use sigma notation to write the sum.
$$\left[2\left(\frac{1}{x}\right)+3\right]+\left[2\left(\frac{3}{8}\right)+3\right]+\ldots+\left[2\left(\frac{5}{8}\right)+3\right]$$

Anurag Kumar

Numerade Educator

Problem 82

Use sigma notation to write the sum.
$$\left[1-\left(\frac{1}{6}\right)^{2}\right]+\left[1-\left(\frac{2}{6}\right)^{2}\right]+\cdots+\left[1-\left(\frac{6}{6}\right)^{2}\right]$$

Anurag Kumar

Numerade Educator

Problem 83

Use sigma notation to write the sum.
$$3-9+27-81+243-729$$

Sheryl Ezze

Numerade Educator

Problem 84

Use sigma notation to write the sum.
$$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots-\frac{1}{128}$$

Anurag Kumar

Numerade Educator

Problem 85

Use sigma notation to write the sum.
$$\frac{1^{2}}{2}+\frac{2^{2}}{6}+\frac{3^{2}}{24}+\frac{4^{2}}{120}+\dots+\frac{7^{2}}{40,320}$$

Anurag Kumar

Numerade Educator

Problem 86

Use sigma notation to write the sum.
$$\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 5}+\dots+\frac{1}{10 \cdot 12}$$

Anurag Kumar

Numerade Educator

Problem 87

Use sigma notation to write the sum.
$$\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}$$

Anurag Kumar

Numerade Educator

Problem 88

Use sigma notation to write the sum.
$$\frac{1}{2}+\frac{2}{4}+\frac{6}{8}+\frac{24}{16}+\frac{120}{32}+\frac{720}{64}$$

Anurag Kumar

Numerade Educator

Problem 89

Find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$$\sum_{i=1}^{\infty}\left(\frac{1}{2}\right)^{i}$$

Anurag Kumar

Numerade Educator

Problem 90

Find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$$\sum_{i=1}^{\infty} 2(\xi)^{i}$$

Anurag Kumar

Numerade Educator

Problem 91

Find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$$\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^{n}$$

Anurag Kumar

Numerade Educator

Problem 92

Find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$$\sum_{n=1}^{\infty} 5\left(-\frac{1}{4}\right)^{n}$$

Anurag Kumar

Numerade Educator

Problem 93

Find the sum of the infinite series.
$$\sum_{i=1}^{\infty} \frac{6}{10}$$

Anurag Kumar

Numerade Educator

Problem 94

Find the sum of the infinite series.
$$\sum_{i=1}^{\infty}\left(\frac{1}{10}\right)^{i}$$

Anurag Kumar

Numerade Educator

Problem 95

Find the sum of the infinite series.
$$\sum_{k=1}^{\infty} 7\left(\frac{1}{10}\right)^{k}$$

Anurag Kumar

Numerade Educator

Problem 96

Find the sum of the infinite series.
$$\sum_{i=1}^{\infty} \frac{2}{10^{i}}$$

Anurag Kumar

Numerade Educator

Problem 97

An investor deposits $\$ 10,000$ in an account that earns $3.5 \%$ interest compounded quarterly. The balance in the account after $n$ quarters is given by $$1-\log _{a} a\left(1+\frac{a_{a} a_{2}}{4}\right)=n-1,23 \ldots$$
(a) Write the first eight terms of the sequence.
(b) Find the balance in the account after 10 years by computing the 40 th term of the sequence.
(c) Is the balance after 20 years twice the balance after
10 years? Explain.

Anurag Kumar

Numerade Educator

Problem 98

The percent $p_{n}$ of United States adults who met federal physical activity guidelines from 2007 through 2014 can be approximated by $$p_{n}=0.0061 n^{3}-0.419 n^{2}+7.85 n+4.9$
$n=7,8, \dots14$$
where $n$ is the year, with $n=7$ corresponding to $2007 .$ (Source:
National Center for Health Statistics)
(a) Write the terms
of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence.
(b) What can you conclude from the bar graph in part (a)?

Bob Bob

Numerade Educator

Problem 99

Determine whether the statement is true or false. Justify your answer.
$$\sum_{l=1}^{4}\left(i^{2}+2 i\right)=\sum_{i=1}^{4} i^{2}+2 \sum_{i=1}^{4} i$$

Yujie Wang

College of San Mateo

Problem 100

Determine whether the statement is true or false. Justify your answer.
$$\sum_{j=1}^{4} 2^{j}=\sum_{j=3}^{6} 2^{j-2}$$

AG

Ankit Gupta

Numerade Educator

Problem 101

Use the following definition of the arithmetic mean $\bar{x}$ of a set of $n$ measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n^{*}}$.
$$\bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$$
Find the arithmetic mean of the six checking account balances $\$ 327.15,$ S785.69, $\$ 433.04, \quad \$ 265.38$
$\$ 604.12,$ and $\$ 590.30 .$ Use the statistical capabilities of a graphing utility to verify your result.

Anurag Kumar

Numerade Educator

Problem 102

Use the following definition of the arithmetic mean $\bar{x}$ of a set of $n$ measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n^{*}}$.
$$\bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$$
Prove that $\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)=0$

Anurag Kumar

Numerade Educator

Problem 103

Proof Prove that
$\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=\sum_{i=1}^{n} x_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} x_{i}\right)^{2}$

Anurag Kumar

Numerade Educator

Problem 104

The graph represents the first 10 terms of a sequence. Complete each expression for the apparent $n$ th term $\left(a_{n}\right)$ of the sequence. Which expressions are appropriate to represent the $\operatorname{cost} a_{n}$ to buy $n$ MP3 songs at a cost of $\$ 1$ per song? Explain.
(a) $a_{n}=1$
(b) $a_{n}=\frac{1}{(n-1) !}$
(c) $a_{n}=\sum_{k=1}^{n}$

Bob Bob

Numerade Educator

Problem 105

Describe the error in finding the sum.
$$\begin{aligned}
\sum_{k=1}^{4}\left(3+2 k^{2}\right) &=\sum_{k=1}^{4} 3+\sum_{k=1}^{4} 2 k^{2} \\
&=3+(2+8+18+32) \\
&=63
\end{aligned}$$

Anurag Kumar

Numerade Educator

Problem 106

Describe the error in finding the sum.
$$\begin{aligned}
\sum_{n=0}^{3}(-1)^{n} n ! &=(-1)(1)+(1)(2)+(-1)(6) \\
&=-5
\end{aligned}$$

Anurag Kumar

Numerade Educator

Problem 107

A $3 \times 3 \times 3$ cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure.
(FIGURE CANNOT COPY)
(a) Determine how many unit cubes of the $3 \times 3 \times 3$ cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces.
(b) Repeat part (a) for a $4 \times 4 \times 4$ cube, a $5 \times 5 \times 5$ cube, and a $6 \times 6 \times 6$ cube.
(c) Write formulas you could use to repeat part (a) for an $n \times n \times n$ cube.

Anurag Kumar

Numerade Educator