Chapter 12

CONCEPT CHECK \(\quad\) When using the method of mathematical induction as stated in this section to prove a statement, the domain of the variable must be all _______.

State a sample space \(S\) with equally likely outcomes for each experiment. A two-headed coin is tossed once.

Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$2,5,8,11, \dots$$

Evaluate each expression. Do not use a calculator. $$4 !$$

CHECKING ANALYTIC SKILLS Write the terms of the geometric sequence that satisfies the given conditions. Do not use a calculator. $$a_{1}=\frac{5}{3}, r=3, n=4$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=4 n+10$$

Suppose that Step 2 in a proof by mathematical induction can be satisfied, but Step 1 cannot. May we conclude that the proof is complete? Explain.

State a sample space \(S\) with equally likely outcomes for each experiment. Two ordinary coins are tossed.

Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$4,10,16,22, \dots$$

Evaluate each expression. Do not use a calculator. $$6 !$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\frac{5 !}{2 ! 3 !}$$

CHECKING ANALYTIC SKILLS Write the terms of the geometric sequence that satisfies the given conditions. Do not use a calculator. $$a_{1}=-\frac{3}{4}, r=\frac{2}{3}, n=4$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=6 n-3$$

CONCEPT CHECK For which positive integers is the statement \(2^{n}>2 n\) not true?

State a sample space \(S\) with equally likely outcomes for each experiment. Three ordinary coins are tossed.

Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$3,-2,-7,-12, \dots$$

Evaluate each expression. Do not use a calculator. $$(4-2) !$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\frac{7 !}{3 ! 4 !}$$

CHECKING ANALYTIC SKILLS Write the terms of the geometric sequence that satisfies the given conditions. Do not use a calculator. $$a_{4}=5, a_{5}=10, n=5$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=3(n-1)+5$$

Write out in full and verify the statements \(S_{1}, S_{2}, S_{3}, S_{4}\) and \(S_{5}\) for the following formula. Then use mathematical induction to prove that the statement is true for every positive integer \(n\) $$ 2+4+6+\dots+2 n=n(n+1) $$

State a sample space \(S\) with equally likely outcomes for each experiment. Five slips of paper, each of which is marked with the number \(1,2,3,4,\) or \(5,\) are placed in a box. After mixing well, two slips are drawn, with the order not important.

Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$-8,-12,-16,-20, \dots$$

Evaluate each expression. Do not use a calculator. $$(5-2) !$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\frac{8 !}{5 ! 3 !}$$

CHECKING ANALYTIC SKILLS Write the terms of the geometric sequence that satisfies the given conditions. Do not use a calculator. $$a_{3}=16, a_{4}=8, n=5$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=-2 n+6$$

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

Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$x+3 y, 2 x+5 y, 3 x+7 y, \dots$$

Evaluate each expression. Do not use a calculator. $$\frac{6 !}{5 !}$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{l} 8 \\\3\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{1}=5, r=-2$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=2^{n-1}$$

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

State a sample space \(S\) with equally likely outcomes for each experiment. A die is rolled and then a coin is tossed.

Find the common difference \(d\) for each arithmetic sequence. Do not use a calculator. $$t^{2}+q,-4 t^{2}+2 q,-9 t^{2}+3 q, \dots$$

Evaluate each expression. Do not use a calculator. $$\frac{7 !}{6 !}$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{l}7 \\\4\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{1}=8, r=-5$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=-3^{n}$$

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

Write the first five terms of each arithmetic sequence. Do not use a calculator. The first term is \(8,\) and the common difference is 6.

Evaluate each expression. Do not use a calculator. $$\frac{8 !}{6 !}$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{c}10 \\\8\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{2}=-4, r=-3$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\left(\frac{1}{3}\right)^{n}(n-1)$$

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

Write each event in set notation. Give the probability of the event. In Exercise 2 A. both coins show the same face. B. at least one coin turns up heads.

Write the first five terms of each arithmetic sequence. Do not use a calculator. The first term is \(-2,\) and the common difference is 12.

Evaluate each expression. Do not use a calculator. $$\frac{9 !}{7 !}$$