🚨 Hurry, space in our FREE summer bootcamps is running out. 🚨Claim your spot here.

College Algebra 12th

R. David Gustafson, Jeff Hughes

Chapter 8

Sequences, Series, and Probability

Educators


Problem 1

Fill in the blanks. In the expansion of a binomial, there will be one more term than the $____$ of the binomial.

Check back soon!

Problem 2

Fill in the blanks. The $____$ of each term in a binomial expansion is the same as the exponent of the binomial.

Check back soon!

Problem 3

Fill in the blanks. The Answer $____$ term in a binomial expansion is the first term raised to the power of the binomial.

Check back soon!

Problem 4

Fill in the blanks. In the expansion of $(a+b)^{n},$ the $\quad\quad$ on $a$ decrease by 1 in each successive term.

Check back soon!

Problem 5

Fill in the blanks. Expand 7!:

Check back soon!

Problem 6

Fill in the blanks. $$0 !=____$$

Check back soon!

Problem 7

Fill in the blanks. $n$____ =n !$$

Check back soon!

Problem 8

Fill in the blanks. In the seventh term of $(a+b)^{11},$ the exponent on $a$ is $____$

Check back soon!

Problem 9

Evaluate each expression. $$5 !$$

Check back soon!

Problem 10

Evaluate each expression. $$-5 !$$

Check back soon!

Problem 11

Evaluate each expression. $$3 ! \cdot 6 !$$

Check back soon!

Problem 12

Evaluate each expression. $$0 ! \cdot 7 !$$

Check back soon!

Problem 13

Evaluate each expression. $$6 !+6 !$$

Check back soon!

Problem 14

Evaluate each expression. $$5 !-2 !$$

Check back soon!

Problem 15

Evaluate each expression. $$\frac{9 !}{12 !}$$

Check back soon!

Problem 16

Evaluate each expression. $$\frac{8 !}{5 !}$$

Check back soon!

Problem 17

Evaluate each expression. $$\frac{5 ! \cdot 7 !}{9 !}$$

Check back soon!

Problem 18

Evaluate each expression. $$\frac{3 ! \cdot 5 ! \cdot 7 !}{1 ! 8 !}$$

Check back soon!

Problem 19

Evaluate each expression. $$\frac{18 !}{6 !(18-6) !}$$

Check back soon!

Problem 20

Evaluate each expression. $$\frac{15 !}{9 !(15-9) !}$$

Check back soon!

Problem 21

Use Pascal's Triangle to expand each binomial. $$(a+b)^{5}$$

Check back soon!

Problem 22

Use Pascal's Triangle to expand each binomial. $$(a+b)^{7}$$

Check back soon!

Problem 23

Use Pascal's Triangle to expand each binomial. $$(x-y)^{3}$$

Check back soon!

Problem 24

Use Pascal's Triangle to expand each binomial. $$(x-y)^{7}$$

Check back soon!

Problem 25

Use the Binomial Theorem to expand each binomial. $$(a+b)^{3}$$

Check back soon!

Problem 26

Use the Binomial Theorem to expand each binomial. $$(a+b)^{4}$$

Check back soon!

Problem 27

Use the Binomial Theorem to expand each binomial. $$(a-b)^{5}$$

Check back soon!

Problem 28

Use the Binomial Theorem to expand each binomial. $$(x-y)^{4}$$

Check back soon!

Problem 29

Use the Binomial Theorem to expand each binomial. $$(2 x+y)^{3}$$

Check back soon!

Problem 30

Use the Binomial Theorem to expand each binomial. $$(x+2 y)^{3}$$

Check back soon!

Problem 31

Use the Binomial Theorem to expand each binomial. $$(x-2 y)^{3}$$

Check back soon!

Problem 32

Use the Binomial Theorem to expand each binomial. $$(2 x-y)^{3}$$

Check back soon!

Problem 33

Use the Binomial Theorem to expand each binomial. $$(2 x+3 y)^{4}$$

Check back soon!

Problem 34

Use the Binomial Theorem to expand each binomial. $$(2 x-3 y)^{4}$$

Check back soon!

Problem 35

Use the Binomial Theorem to expand each binomial. $$(x-2 y)^{4}$$

Check back soon!

Problem 36

Use the Binomial Theorem to expand each binomial. $$(x+2 y)^{4}$$

Check back soon!

Problem 37

Use the Binomial Theorem to expand each binomial. $$(x-3 y)^{5}$$

Check back soon!

Problem 38

Use the Binomial Theorem to expand each binomial. $$(3 x-y)^{5}$$

Check back soon!

Problem 39

Use the Binomial Theorem to expand each binomial. $$\left(\frac{x}{2}+y\right)^{4}$$

Check back soon!

Problem 40

Use the Binomial Theorem to expand each binomial. $$\left(x+\frac{y}{2}\right)^{4}$$

Check back soon!

Problem 41

Find the required term in each binomial expansion. $$(a+b)^{4} ; 3 \mathrm{rd} \text { term }$$

Check back soon!

Problem 42

Find the required term in each binomial expansion. $$(a-b)^{4} ; 2 n d \text { term }$$

Check back soon!

Problem 43

Find the required term in each binomial expansion. $$(a+b)^{7} ; 5 \text { th term }$$

Check back soon!

Problem 44

Find the required term in each binomial expansion. $$(a+b)^{5} ; 4 \text { th term }$$

Check back soon!

Problem 45

Find the required term in each binomial expansion. $$(a-b)^{5} ; 6 \text { th term }$$

Check back soon!

Problem 46

Find the required term in each binomial expansion. $$(a-b)^{8} ; 7 \text { th term }$$

Check back soon!

Problem 47

Find the required term in each binomial expansion. $$(a+b)^{17} ; 5 \text { th term }$$

Check back soon!

Problem 48

Find the required term in each binomial expansion. $$(a-b)^{12} ; 3 r d \text { term }$$

Check back soon!

Problem 49

Find the required term in each binomial expansion. $$(a-\sqrt{2})^{4} ; 2 n d \text { term }$$

Check back soon!

Problem 50

Find the required term in each binomial expansion. $$(a-\sqrt{3})^{8} ; 3 \mathrm{rd} \text { term }$$

Check back soon!

Problem 51

Find the required term in each binomial expansion. $$(a+\sqrt{3} b)^{9} ; 5 \text { th term }$$

Check back soon!

Problem 52

Find the required term in each binomial expansion. $$(\sqrt{2} a-b)^{7} ; 4 \text { th term }$$

Check back soon!

Problem 53

Find the required term in each binomial expansion. $$\left(\frac{x}{2}+y\right)^{4} ; 3 r d \text { term }$$

Check back soon!

Problem 54

Find the required term in each binomial expansion. $$\left(m+\frac{n}{2}\right)^{8} ; 3 \mathrm{rd} \text { term }$$

Check back soon!

Problem 55

Find the required term in each binomial expansion. $$\left(\frac{r}{2}-\frac{s}{2}\right)^{11} ; 10 \text { th term }$$

Check back soon!

Problem 56

Find the required term in each binomial expansion. $$\left(\frac{p}{2}-\frac{q}{2}\right)^{9} ; 6 \text { th term }$$

Check back soon!

Problem 57

Find the required term in each binomial expansion. $$(a+b)^{n} ; 4 \text { th term }$$

Check back soon!

Problem 58

Find the required term in each binomial expansion. $$(a-b)^{n} ; 5 \text { th term }$$

Check back soon!

Problem 59

Find the required term in each binomial expansion. $$(a+b)^{n} ; r t h \text { term }$$

Check back soon!

Problem 60

Find the required term in each binomial expansion. $$(a+b)^{n} ;(r+1) \text { th term }$$

Check back soon!

Problem 61

Describe how to construct Pascal's Triangle.

Check back soon!

Problem 62

What is binomial expansion?

Check back soon!

Problem 63

Explain why the terms alternate in the binomial expansion of $(x-y)^{8}$

Check back soon!

Problem 64

Define factorial notation and explain how to evaluate $10 !$

Check back soon!

Problem 65

With a calculator, evaluate $69 ! .$ Explain why we cannot find $70 !$ with a calculator.

Check back soon!

Problem 66

Find the sum of the numbers in each row of the first ten rows of Pascal's Triangle. Do you see a pattern?

Check back soon!

Problem 67

Show that the sum of the coefficients in the binomial expansion of $(x+y)^{n}$ is $2^{n} .$ (Hint: Let
$x=y=1 .)$

Check back soon!

Problem 68

Explain how the $r$ th term of a binomial expansion is constructed.

Check back soon!

Problem 69

If we applied the pattern of coefficients to the coefficient of the first term in the Binomial Theorem, it would be $\frac{n !}{0 !(n-0) !} .$ Show that this expression equals 1

Check back soon!

Problem 70

If we applied the pattern of coefficients to the coefficient of the last term in the Binomial Theorem, it would be $\frac{n !}{n !(n-n) !}$. Show that this expression equals 1 .

Check back soon!

Problem 71

Determine if the statement is true or false. If the statement is false, then correct it and make it true. $$0 !=0$$

Check back soon!

Problem 72

Determine if the statement is true or false. If the statement is false, then correct it and make it true. The first term in the expansion of $(a+b)^{999}$ is $a^{999}$

Check back soon!

Problem 73

Determine if the statement is true or false. If the statement is false, then correct it and make it true. The last term in the expansion of $(a-b)^{888}$ is $b^{888}$

Check back soon!

Problem 74

Determine if the statement is true or false. If the statement is false, then correct it and make it true. For the expansion of $(a+b)^{n \pi}$, the exponents on $a$ increase by 1 in each successive term.

Check back soon!

Problem 75

Determine if the statement is true or false. If the statement is false, then correct it and make it true. For the expansion of $(a-b)^{666},$ the exponents on $b$ decrease by 1 in each successive term.

Check back soon!

Problem 76

Determine if the statement is true or false. If the statement is false, then correct it and make it true. For the expansion of $(a-b)^{\text {cos }},$ the exponents on $b$ decrease by 1 in each successive term.

Check back soon!

Problem 77

Determine if the statement is true or false. If the statement is false, then correct it and make it true. The number of terms in the binomial expansion of $(a-b)^{444}$ is 444

Check back soon!

Problem 78

Determine if the statement is true or false. If the statement is false, then correct it and make it true. To find the binomial expansion of $\left(x^{333}-y^{222}\right)^{111},$ it is helpful to rewrite the expression inside the parentheses as $x^{333}+\left(-y^{222}\right)$

Check back soon!

Problem 79

Determine if the statement is true or false. If the statement is false, then correct it and make it true. The constant term in the expansion of $\left(a-\frac{1}{a}\right)^{10}$ is 252

Check back soon!

Problem 80

Determine if the statement is true or false. If the statement is false, then correct it and make it true. The coefficient of $x^{5}$ in the expansion of $\left(x-\frac{1}{x}\right)^{9}$ is 36

Check back soon!