Chapter 10

\(\sum_{k=1}^{4}\left(k^{2}-5\right) \quad \)

Find the sum of the infinite geometric series if it exists. $$1+\frac{3}{2}+\frac{9}{4}+\frac{27}{8}+\cdots$$

How many 13 -card hands dealt from a standard deck will have exactly seven spades?

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(3 x^{2}-\sqrt{y}\right)^{9}, \quad \text { fifth term } $$

\(\sum_{k=1}^{10}\left[1+(-1)^{k}\right]\)

Find the sum of the infinite geometric series if it exists. $$256+192+144+108+\cdots$$

Card and die experiment Each suit in a deck is made up of an ace (A), nine numbered cards \((2,3, \ldots, 10)\), and three face cards (J, Q, K). An experiment consists of drawing a single card from a deck followed by rolling a single die. (a) Describe the sample space \(S\) of the experiment, and find \(n(S)\). (b) Let \(E_{1}\) be the event consisting of the outcomes in which a numbered card is drawn and the number of dots on the die is the same as the number on the card. Find \(n\left(E_{1}\right), n\left(E_{1}^{\prime}\right)\), and \(P\left(E_{1}\right)\). (c) Let \(E_{2}\) be the event in which the card drawn is a face card, and let \(E_{3}\) be the event in which the number of dots on the die is even. Are \(E_{2}\) and \(E_{3}\) mutually exclusive? Are they independent? Find \(P\left(E_{2}\right), P\left(E_{3}\right)\), \(P\left(E_{2} \cap E_{3}\right)\), and \(P\left(E_{2} \cup E_{3}\right)\). (d) Are \(E_{1}\) and \(E_{2}\) mutually exclusive? Are they independent? Find \(P\left(E_{1} \cap E_{2}\right)\) and \(P\left(E_{1} \cup E_{2}\right)\).

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(\frac{1}{3} u+4 v\right)^{8} ; \quad \text { seventh term } $$

Exer. 37-38: (a) Evaluate the given formula for the stated values of \(n\), and solve the resulting system of equations for \(a, b, c\), and \(d\). (This method can sometimes be used to obtain formulas for sums.) (b) Compare the result in part (a) with the indicated exercise, and explain why this method does not prove that the formula is true for every \(n\). $$ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=a n^{3}+b n^{2}+c n ; $$ \(n=1,2,3\) (Exercise 9)

Selecting theater seats Three married couples have purchased tickets for a play. Spouses are to be seated next to each other, and the six seats are in a row. In how many ways can the six people be seated?

\(\sum_{k=0}^{5} k(k-2) \quad\)

Find the sum of the infinite geometric series if it exists. $$250-100+40-16+\cdots$$

Letter and number experiment An experiment consists of selecting a letter from the alphabet and one of the digits 0 , \(1, \ldots, 9\). (a) Describe the sample space \(S\) of the experiment, and find \(n(S)\). (b) Suppose the letters of the alphabet are assigned numbers as follows: \(A=1, B=2, \ldots, Z=26\). Let \(E_{1}\) be the event in which the units digit of the number assigned to the letter of the alphabet is the same as the digit selected. Find \(n\left(E_{1}\right), n\left(E_{1}^{\prime}\right)\), and \(P\left(E_{1}\right)\). (c) Let \(E_{2}\) be the event that the letter is one of the five vowels and \(E_{3}\) the event that the digit is a prime number. Are \(E_{2}\) and \(E_{3}\) mutually exclusive? Are they independent? Find \(P\left(E_{2}\right), P\left(E_{3}\right), P\left(E_{2} \cap E_{3}\right)\), and \(P\left(E_{2} \cup E_{3}\right)\). (d) Let \(E_{4}\) be the event that the numerical value of the letter is even. Are \(E_{2}\) and \(E_{4}\) mutually exclusive? Are they independent? Find \(P\left(E_{2} \cap E_{4}\right)\) and \(P\left(E_{2} \cup E_{4}\right)\).

Without expanding completely, find the indicated term(s) in the expansion of the expression. \(\left(3 x^{2}-y^{3}\right)^{10}\) fourth term

Exer. 37-38: (a) Evaluate the given formula for the stated values of \(n\), and solve the resulting system of equations for \(a, b, c\), and \(d\). (This method can sometimes be used to obtain formulas for sums.) (b) Compare the result in part (a) with the indicated exercise, and explain why this method does not prove that the formula is true for every \(n\). $$ \begin{aligned} &1^{3}+2^{3}+3^{3}+\cdots+n^{3}=a n^{4}+b n^{3}+c n^{2}+d n \\ &n=1,2,3,4 \text { (Exercise } 10) \end{aligned} $$

Horserace results Ten horses are entered in a race. If the possibility of a tie for any place is ignored, in how many ways can the first-, second-, and third-place winners be determined?

Exer. 37-40: Find the number of terms in the arithmetic sequence with the given conditions. $$ a_{1}=-1, \quad d=\frac{1}{5}, \quad S=21 $$

\(\sum_{k=0}^{4}(k-1)(k-3)\)

Tossing dice If two dice are tossed, find the probability that the sum is greater than 5 .

Find the rational number represented by the repeating decimal. $$0 . \overline{23}$$

Show that \(C(n, r-1)+C(n, r)=C(n+1, r)\). Interpret this formula in terms of Pascal's triangle.

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(x^{1 / 2}+y^{1 / 2}\right)^{8}, \quad \text { middle term } $$

Exer. 39-42: Prove that the statement is true for every positive integer \(n\). \(\sin (\theta+n \pi)=(-1)^{n} \sin \theta\)

Lunch possibilities Owners of a restaurant advertise that they offer \(1,114,095\) different lunches based on the fact that they have 16 "free fixins" to go along with any of their 17 menu items (sandwiches, hot dogs, and salads). How did they arrive at that number?

Exer. 37-40: Find the number of terms in the arithmetic sequence with the given conditions. $$ a_{1}=-\frac{29}{6}, \quad d=\frac{1}{3}, \quad S=-36 $$

\(\sum_{k=3}^{6} \frac{k-5}{k-1}\) $$

Tossing dice If three dice are tossed, find the probability that the sum is less than 16 .

Find the rational number represented by the repeating decimal. $$0.0 \overline{71}$$

Without expanding completely, find the indicated term(s) in the expansion of the expression. \(\left(r s^{2}+t\right)^{7}\) two middle terms

Exer. 39-42: Prove that the statement is true for every positive integer \(n\). \(\cos (\theta+n \pi)=(-1)^{n} \cos \theta\)

Shuffling cards (a) In how many ways can a standard deck of 52 cards be shuffled? (b) In how many ways can the cards be shuffled so that the four aces appear on the top of the deck?

$\sum_{k=1}^{6} \frac{3}{k+1} $$

Find the rational number represented by the repeating decimal. $$2.4 \overline{17}$$

Family makeup Assuming that girl-boy births are equally probable, find the probability that a family with five children has (a) all boys (b) at least one girl

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(2 y+x^{2}\right)^{8} ; \quad \text { term that contains } x^{10} $$

Exer. 39-42: Prove that the statement is true for every positive integer \(n\). Prove De Moivre's theorem: $$ [r(\cos \theta+i \sin \theta)]^{n}=r^{n}(\cos n \theta+i \sin n \theta) $$ for every positive integer \(n\).

Numerical patindromes A palindrome is an integer, such as 45654 , that reads the same backward and forward. (a) How many five-digit palindromes are there? (b) How many \(n\)-digit palindromes are there?

Insert five arithmetic means between 2 and \(10 .\)

\(\sum_{k=1}^{5}(-3)^{k-1} \quad\)

Slot machine A standard slot machine contains three reels, and each reel contains 20 symbols. If the first reel has five bells, the middle reel four bells, and the last reel two bells, find the probability of obtaining three bells in a row.

Find the rational number represented by the repeating decimal. $$10 . \overline{5}$$

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(x^{2}-2 y^{3}\right)^{5} ; \quad \text { term that contains } y^{6} $$

Exer. 39-42: Prove that the statement is true for every positive integer \(n\). Prove that for every positive integer \(n \geq 3\), the sum of the interior angles of an \(n\)-sided polygon is given by the expression \((n-2) \cdot 180^{\circ}\).

Insert three arithmetic means between 3 and \(-5\).

\(\sum_{k=0}^{4} 3\left(2^{k}\right)\)

ESP experiment In a simple experiment designed to test ESP, four cards (jack, queen, king, and ace) are shuffled and then placed face down on a table. The subject then attempts to identify each of the four cards, giving a different name to each of the cards. If the individual is guessing, find the probability of correctly identifying (a) all four cards (b) exactly two of the four cards

Find the rational number represented by the repeating decimal. $$5 \longdiv { 1 4 6 }$$

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(3 y^{3}-2 x^{2}\right)^{4} ; \quad \text { term that contains } y^{9} $$

The graph of $$ y=\frac{x ! e^{x}}{x^{x} \sqrt{2 \pi x}} $$ has a horizontal asymptote of \(y=1\). Use this fact to find an approximation for \(n !\) if \(n\) is a large positive integer.

(a) Find the number of integers between 32 and 395 that are divisible by 6 . (b) Find their sum.