Chapter 8

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

Evaluate each expression. $$C(16,3)$$

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

Find \(a_{1}\) for each arithmetic sequence. $$a_{5}=27, a_{15}=87$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$a_{1}=8.423, r=2.859$$

The following table shows the probability that a customer at a department store will make a purchase in the indicated price range.$$\begin{array}{|l|c|} \hline \multicolumn{1}{|c|}\text { cost } & \text { Probability } \\ \hline \text { Below \$5 } & 0.25 \\ \$ 5-\$ 19.99 & 0.37 \\ \$ 20-\$ 39.99 & 0.11 \\ \$ 40-\$ 69.99 & 0.09 \\ \$ 70-\$ 99.99 & 0.07 \\ \$ 100-\$ 149.99 & 0.08 \\ \$ 150 \text { or more } & 0.03 \end{array}$$ Find the probability that a customer makes a purchase that is (a) less than \(\$ 20\) (b) \(\$ 40\) or more. (c) more than \(\$ 99.99\) (d) less than \(\$ 100\).

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

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

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

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

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

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$a_{1}=-3.772, r=-1.553$$

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 \(\$ 5000 ?\)

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

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

Find the first four terms of each sequence. $$a_{1}=0, a_{n}=3+n \cdot a_{n-1}, \text { for } n>1$$

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

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

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

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

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

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

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

The law firm of Alam, Bartolini, Chinn, Dickinson, and Ellsberg has two senior partners: Alam and Bartolini. Two of the attorneys 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.

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

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

Use a calculator to evaluate each expression. $$32 P_{4}$$

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

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

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

Prove each statement by mathematical induction. $$\text { If } 0

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

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

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

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$\sum_{j=1}^{5} 243\left(\frac{2}{3}\right)^{j}$$

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

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

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

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

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

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

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$\sum_{k=4}^{10}(-2)^{k}$$

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. Exactly 2 girls and 3 boys

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

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

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

Find the sum for each series. $$\sum_{j=1}^{4} \frac{1}{j}$$

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

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. Exactly 3 girls and 2 boys

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