Chapter 12

Use a calculator to evaluate each expression. \({ }_{20} P_{5}\)

Write the binomial expansion for each expression. $$(p-q)^{5}$$

Find the first four terms of each sequence. \(a_{1}=1, a_{2}=1, a_{n}=a_{n-1}+a_{n-2},\) for \(n \geq 3\) (the Fibonacci sequence)

Work each problem. State Lottery One game in a state lottery requires you to pick 1 heart, 1 club, 1 diamond, and 1 spade, in that order, from the 13 cards in each suit. What is the probability of getting all four picks correct and winning \(\$ 5000 ?\)

Prove each statement by mathematical induction. \((a b)^{n}=a^{n} b^{n}\) (Assume that \(a\) and \(b\) are constant.)

Find \(a_{1}\) for each arithmetic sequence. $$a_{12}=60, a_{20}=84$$

Use a calculator to evaluate each expression. \({ }_{100} P_{5}\)

Write the binomial expansion for each expression. $$(a-b)^{7}$$

Find the first four terms of each sequence. \(a_{1}=1, a_{2}=3, a_{n}=a_{n-1}+a_{n-2},\) for \(n \geq 3\) (The Lucas sequence)

Prove each statement by mathematical induction. \(2^{n}>2 n,\) if \(n \geq 3\)

Find \(a_{1}\) for each arithmetic sequence. $$a_{5}=-3, a_{18}=-29$$

Use a calculator to evaluate each expression. $$_{15} P_{8}$$

Write the binomial expansion for each expression. $$\left(r^{2}+s\right)^{5}$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 29 and 30 to the nearest hundredih. $$18,-9, \frac{9}{2},-\frac{9}{4}, \dots$$

Find the first four terms of each sequence. $$a_{1}=5, a_{n}=3 n+3 a_{n-1}, \text { for } n>1$$

Work each problem. Partner Selection \(\quad\) The law firm of Alam, Bartolini, Chinn, Dickinson, and Ellsberg has two senior partners: Alam and Bartolini. Two of the attomeys are to be selected to attend a conference. Assuming that all are equally likely to be selected, find each probability. A. Chinn is selected. B. Alam and Dickinson are selected. C. At least one senior partner is selected.

Prove each statement by mathematical induction. \(3^{n}>2 n+1,\) if \(n \geq 2\)

Find \(a_{1}\) for each arithmetic sequence. $$a_{6}=-8, a_{7}=-18$$

Write the binomial expansion for each expression. $$\left(m+n^{2}\right)^{4}$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 29 and 30 to the nearest hundredih. $$12,-4, \frac{4}{3},-\frac{4}{9}, \dots$$

Find the first four terms of each sequence. $$a_{1}=2, a_{n}=n \cdot a_{n-1}, \text { for } n>1$$

Prove each statement by mathematical induction. If \(a>1,\) then \(a^{n}>1\)

Find \(a_{1}\) for each arithmetic sequence. $$S_{3}=75, a_{3}=22$$

Use a calculator to evaluate each expression. $$_{20} C_{5}$$

Write the binomial expansion for each expression. $$(p+2 q)^{4}$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 29 and 30 to the nearest hundredih. $$a_{1}=8.423, r=2.859$$

Find the first four terms of each sequence. $$a_{1}=2, a_{2}=3, a_{n}=a_{n-1} \cdot a_{n-2} \text { for } n>2$$

Work each problem. Explain why the probability of an event must be a number between 0 and 1 inclusive.

Prove each statement by mathematical induction. If \(a>1,\) then \(a^{n}>a^{n-1}\)

Find \(a_{1}\) for each arithmetic sequence. $$S_{20}=-1300, a_{20}=-122$$

Use a calculator to evaluate each expression. $$100 C_{5}$$

Write the binomial expansion for each expression. $$(3 r-s)^{6}$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 29 and 30 to the nearest hundredih. $$a_{1}=-3.772, r=-1.553$$

Find the first four terms of each sequence. $$a_{1}=2, a_{2}=1, a_{n}=2 a_{n-1}^{2}+a_{n-2}, \text { for } n>2$$

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) Exactly 2 girls and 3 boys

Find \(a_{1}\) for each arithmetic sequence. $$S_{16}=-160, a_{16}=-25$$

Prove each statement by mathematical induction. If \(0

Use a calculator to evaluate each expression. $$ { }_{15} C_{8} $$

Write the binomial expansion for each expression. $$(7 p+2 q)^{4}$$

Use a formula to find the sum of each series. $$\sum_{i=1}^{5} 3^{i}$$

Find the sum for each series. $$\sum_{i=1}^{5}(2 i+1)$$

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) Exactly 3 girls and 2 boys

Prove each statement by mathematical induction. \(2^{n}>n^{2},\) for \(n>4\)

Find \(a_{1}\) for each arithmetic sequence. $$S_{28}=2926, a_{28}=199$$

Use a calculator to evaluate each expression. \({ }_{32} C_{4}\)

Write the binomial expansion for each expression. $$(4 a-5 b)^{5}$$

Use a formula to find the sum of each series. $$\sum_{i=1}^{4} 2^{i}$$

Find the sum for each series. $$\sum_{i=1}^{6}(3 i-2)$$

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) No girls

Prove each statement by mathematical induction. If \(n \geq 4,\) then \(n !>2^{n}\)