Chapter 11

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (c+2)^{5} $$

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

Evaluate each expression. \(1-\frac{_{3} P_{2}}{_{4} P_{3}}\)

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 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. $$ 1,5,9,13, \dots $$

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 \(\\{M M M, M M F, M F M, M F F, F M M FMF, FFM, FFF\)} - Find the probability of selecting a family with $$\text{at least two female children.}$$

In Exercises \(17-24,\) 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. $$ 0.0007,-0.007,0.07,-0.7, \ldots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (c+3)^{5} $$

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

Evaluate each expression. \(1-\frac{_{5} P_{3}}{_{10} P_{4}}\)

Use mathematical induction to prove that each statement is true for every positive integer n. \(\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots+\frac{1}{(n+1)(n+2)}=\frac{n}{2 n+4}\)

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. $$ 2,7,12,17, \dots $$

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,

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x-1)^{5} $$

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

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

Use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of \(n^{2}-n\)

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. $$ 7,3,-1,-5, \dots $$

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: \(3,6,12,24, \dots\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x-2)^{5} $$

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

Evaluate each expression. \(\frac{_{10} C_{3}}{_{6} C_{4}}-\frac{46 !}{44 !}\)

Use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of \(n^{2}+3 n\)

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. $$ 6,1,-4,-9, \dots $$

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins the jackpot by matching all five numbers drawn from white balls \((1 \text { through } 56\) ) and matching the number on the gold Mega Ball \((1 \text { through } 46) .\) What is the probability of winning the jackpot?

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: \(3,-6,12,-24, \dots\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (3 x-y)^{5} $$

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

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

Use mathematical induction to prove that each statement is true for every positive integer n. 6 is a factor of \(n(n+1)(n+2)\)

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. $$ a_{1}=9, d=2 $$

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins a minimum award of \(\$ 10,000\) by correctly matching four numbers drawn from white balls ( 1 through 56 ) and matching the number on the gold Mega Bali (1 through 46 ). What is the probability of winning this consolation prize?

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: \(4,-12,36,-108, \dots\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x-3 y)^{5} $$

evaluate each factorial expression. $$ \frac{(2 n+1) !}{(2 n) !} $$

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

Use mathematical induction to prove that each statement is true for every positive integer n. 3 is a factor of \(n(n+1)(n-1)\)

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. $$ a_{1}=6, d=3 $$

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins a minimum award of \(\$ 150\) by correctly matching three numbers drawn from white balls \((1 \text { through } 56\) ) and matching the number on the gold Mega Ball \((1 \text { through } 46)\) What is the probability of winning this consolation prize?

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 14 terms of the geometric sequence: \(-\frac{3}{2}, 3,-6,12, \dots\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (2 a+b)^{6} $$

find each indicated sum. $$ \sum_{i=1}^{6} 5 i $$

The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or station wagon). In how many ways can you order the car?

Use mathematical induction to prove that each statement is true for every positive integer n. \(\sum_{i=1}^{n} 5 \cdot 6^{i}=6\left(6^{n}-1\right)\)

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. $$ a_{1}=-20, d=-4 $$

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins a minimum award of \(\$ 10\) by correctly matching two numbers drawn from white balls \((1 \text { through } 56\) ) and matching the number on the gold Mega Ball \((1 \text { through } 46)\) What is the probability of winning this consolation prize?

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 14 terms of the geometric sequence: \(-\frac{1}{24}, \frac{1}{12},-\frac{1}{6}, \frac{1}{3}, \ldots\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (a+2 b)^{6} $$

find each indicated sum. $$ \sum_{i=1}^{6} 7 i $$

A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?