Chapter 8

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(y-4)^{4}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$$

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$12,6,3, \frac{3}{2}, \dots$$

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(|H H, H T, T H, T T| .\) Find the probability of getting two heads.

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{60}\) when \(a_{1}=35, d=-3\)

Evaluate each expression. $$ \frac{_7{P_{3}}}{3 !}-_{7}C_{3} $$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$\left(2 x^{3}-1\right)^{4}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$1 \cdot 2+2 \cdot 3+3 \cdot 4+\dots+n(n+1)=\frac{n(n+1)(n+2)}{3}$$

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$1.5,-3,6,-12, \dots$$

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(|H H, H T, T H, T T| .\) Find the probability of getting the same outcome on each toss.

Evaluate each expression. $$ \frac{_{20} P_{2}}{2 !}-_{20} C_{2} $$

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{70}\) when \(a_{1}=-32, d=4\)

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$\left(2 x^{5}-1\right)^{4}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$1 \cdot 3+2 \cdot 4+3 \cdot 5+\dots+n(n+2)=\frac{n(n+1)(2 n+7)}{6}$$

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$5,-1, \frac{1}{5},-\frac{1}{25}, \dots$$

The general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$a_{n}--2(n-1) !$$

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{\mathrm{MMM},\) MMF. MFM. MFF. FMM. FMF, FFM, FFF . Find the probability of selecting a family with at least one male child.

Evaluate each expression. $$ 1-\frac{_3 P_{2}}{_3P_{3}} $$

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$1,5,9,13, \dots$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(c+2)^{5}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$0.0004,-0.004,0.04,-0.4, \dots$$

Evaluate each factorial expression. $$\frac{17 !}{15 !}$$

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{\mathrm{MMM},\) MMF. MFM. MFF. FMM. FMF, FFM, FFF . Find the probability of selecting a family with at least two female children.

Evaluate each expression. $$ 1-\frac{_5 P_{3}}{_{10} P_{4}} $$

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$2,7,12,17, \dots$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(c+3)^{5}$$

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$0.0007,-0.007,0.07,-0.7, \ldots$$

Evaluate each factorial expression. $$\frac{18 !}{16 !}$$

Evaluate each expression. $$ \frac{_{7} C_{3}}{_{5} C_{4}}-\frac{98 !}{96 !} $$

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$7,3,-1,-5, \dots$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-1)^{3}$$

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) 2 is a factor of \(n^{2}-n\).

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 12 terms of the geometric sequence: $$2,6,18,54, \dots$$

Evaluate each factorial expression. $$\frac{16 !}{2 ! 14 !}$$

Evaluate each expression. $$ \frac{_{10} C_{3}}{_6 C_{4}}-\frac{46 !}{44 !} $$

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$6,1,-4,-9, \dots$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-2)^{5}$$

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 12 terms of the geometric sequence: $$3,6,12,24$$

Evaluate each factorial expression. $$\frac{20 !}{2!18 !}$$

To play the California lottery, a person has to select 6 out of 51 numbers, paying \(\$ 1\) for each six-number selection. If you pick six numbers that are the same as the ones drawn by the lottery, you win mountains of money. What is the probability that a person with one combination of six numbers will win? What is the probability of winning if 100 different lottery tickets are purchased?

Evaluate each expression. $$ \frac{_{4} C_{2} \cdot_{6} C_{1}}{_{18} C_{3}} $$

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=9, d=2$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(3 x-y)^{5}$$

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) 6 is a factor of \(n(n+1)(n+2)\).

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 11 terms of the geometric sequence: $$3,-6,12,-24, \dots$$

Evaluate each factorial expression. $$\frac{(n+2) !}{n !}$$

A state lottery is designed so that a player chooses six numbers from 1 to 30 on one lottery ticket. What is the probability that a player with one lottery ticket will win? What is the probability of winning if 100 different lottery tickets are purchased?

Evaluate each expression. $$ \frac{_{5} C_{1} \cdot_{7} C_{2}}{_{12} C_{3}} $$

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=6, d=3$$