Chapter 8

Does the number represent a probability? $$ \frac{11}{13} $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 3+6+9+\dots+3 n=\frac{3 n(n+1)}{2} $$

Find the first four terms of the sequence. \(a_{n}=2 n+1\)

Count the number of ways that the questions on an exam could be answered. Ten true-false questions

Write a sequence whose terms represent the first six positive even integers.

Evaluate the expression. $$ \left(\begin{array}{l} 6 \\ 2 \end{array}\right) $$

Does the number represent a probability? $$ 0.995 $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 1+3+5+\dots+(2 n-1)=n^{2} $$

Find the first four terms of the sequence. \(a_{n}=3(n-1)+5\)

Count the number of ways that the questions on an exam could be answered. Ten multiple-choice questions with five choices each

Write a sequence whose terms represent the first seven positive odd integers.

Evaluate the expression. $$ \left(\begin{array}{l} 4 \\ 0 \end{array}\right) $$

Does the number represent a probability? $$ 2.5 $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 5+10+15+\dots+5 n=\frac{5 n(n+1)}{2} $$

Find the first four terms of the sequence. \(a_{n}=4(-2)^{n-1}\)

Count the number of ways that the questions on an exam could be answered. Five true-false questions and ten multiple-choice questions with four choices each

Write a series that represents the sum of the first six positive even integers. Find its sum.

Does the number represent a probability? $$ 1 $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 4+7+10+\dots+(3 n+1)=\frac{n(3 n+5)}{2} $$

Find the first four terms of the sequence. \(a_{n}=2(3)^{n}\)

Count the number of ways that the questions on an exam could be answered. One question involving matching ten items in one column with ten items in another column, using a one-toone correspondence

Write a series that represents the sum of the first seven positive odd integers, Find its sum.

Does the number represent a probability? $$ 0 $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 3+3^{2}+3^{3}+\dots+3^{n}=\frac{3\left(3^{n}-1\right)}{2} $$

Find the first four terms of the sequence. \(a_{n}=\frac{n}{n^{2}+1}\)

License Plates Count the number of possible license plates with the given constraints. Three digits followed by three letters

Let \(A_{n}\) represent the number of \(U . S .\) AIDS deaths reported \(n\) years after 2000 . Write a series whose sum gives the cumulative number of AIDS deaths from 2005 to 2009 .

Does the number represent a probability? $$ 110 \% $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6} $$

Find the first four terms of the sequence. \(a_{n}=5-\frac{1}{n^{2}}\)

License Plates Count the number of possible license plates with the given constraints. Two letters followed by four digits

Does the number represent a probability? $$ -0.375 $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 1^{3}+2^{3}+3^{3}+\dots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} $$

Find the first four terms of the sequence. \(a_{n}=(-1)^{n}\left(\frac{1}{2}\right)^{n}\)

License Plates Count the number of possible license plates with the given constraints. Three letters followed by three digits or letters

Evaluate the expression. $$ \left(\begin{array}{l} 5 \\ 2 \end{array}\right) $$

Does the number represent a probability? $$ \frac{9}{8} $$

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 5 \cdot 6+5 \cdot 6^{2}+5 \cdot 6^{3}+\dots+5 \cdot 6^{n}=6\left(6^{n}-1\right) $$

Find the first four terms of the sequence. \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\)

License Plates Count the number of possible license plates with the given constraints. Two letters followed by either three or four digits

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. Three \(a\) 's, two \(b\) 's.

Find the probability of each event. Tossing a head with a fair coin

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{n(n+1)}-\frac{n}{n+1} $$

Find the first four terms of the sequence. \(a_{n}=(-1)^{n-1}\left(\frac{2^{n}}{1+2^{n}}\right)\)

Counting Strings Count the number of five-letter strings that can be formed with the given letters, assuming a letter can be used more than once. \(A, B, C\)

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{n}=2 n-1 $$

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. Five \(a^{\prime} s,\) three \(b^{\prime} s\)

Find the probability of each event. Tossing a tail with a fair coin

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ 7 \cdot 8+7 \cdot 8^{2}+7 \cdot 8^{3}+\dots+7 \cdot 8^{n}=8\left(8^{n}-1\right) $$

Find the first four terms of the sequence. \(a_{n}=(-1)^{n-1}\left(\frac{1}{3^{n}}\right)\)