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Exercícios de sequências, contagem e probabilidade · Physics Playground
Exercícios
Matemática
Sequências, contagem e probabilidade
45 problemas com soluções resolvidas, de Algebra and Trigonometry.
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Algebra and Trigonometry
13.1 Sequences and Their Notations
Exemplo 1
Writing the Terms of a Sequence Defined by an Explicit Formula
Write the first five terms of the sequence defined by the explicit formula an =−3n+8.
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Exemplo 2
Writing the Terms of an Alternating Sequence Defined by an Explicit Formula
Write the first five terms of the sequence. an =((−1) ^n n^2)/(n+1)
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Exemplo 3
Writing the Terms of a Sequence Defined by a Piecewise Explicit Formula
Write the first six terms of the sequence. an ={ n^2 if nis not divisible by 3 n/3 if nis divisible by 3
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Exemplo 4
Writing an Explicit Formula for the nth Term of a Sequence
Write an explicit formula for the nth term of each sequence. ⓐ { −2/(11),3/(13),−4/(15),5/(17),−6/(19),… } ⓑ {−2/(25),−2/(125),−2/(625),−2/(3,125),−2/(15,625),…} ⓒ {e^4,e^5,e^6,e^7,e^8,…}
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Exemplo 5
Writing the Terms of a Sequence Defined by a Recursive Formula
Write the first five terms of the sequence defined by the recursive formula. a1 =9 an =3an−1 −20, for n≥2
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Exemplo 6
Writing the Terms of a Sequence Defined by a Recursive Formula
Write the first six terms of the sequence defined by the recursive formula. a1 =1 a2 =2 an =3an−1 +4an−2, for n≥3
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Exemplo 7
Writing the Terms of a Sequence Using Factorials
Write the first five terms of the sequence defined by the explicit formula an =(5n)/((n+2)!).
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Algebra and Trigonometry
13.2 Arithmetic Sequences
Exemplo 1
Finding Common Differences
Is each sequence arithmetic? If so, find the common difference. ⓐ {1,2,4,8,16,...} ⓑ {−3,1,5,9,13,...}
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Exemplo 2
Writing Terms of Arithmetic Sequences
Write the first five terms of the arithmetic sequence with a1 =17 and d=−3.
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Exemplo 3
Writing Terms of Arithmetic Sequences
Given a1 =8 and a4 =14, find a5.
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Exemplo 4
Writing a Recursive Formula for an Arithmetic Sequence
Write a recursive formula for the arithmetic sequence. {−18, −7, 4, 15, 26, …}
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Exemplo 5
Writing the nth Term Explicit Formula for an Arithmetic Sequence
Write an explicit formula for the arithmetic sequence. {2, 12, 22, 32, 42, …}
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Exemplo 6
Finding the Number of Terms in a Finite Arithmetic Sequence
Find the number of terms in the finite arithmetic sequence. {8, 1, –6,..., –41}
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Exemplo 7
Solving Application Problems with Arithmetic Sequences
A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week. ⓐWrite a formula for the child’s weekly allowance in a given year. ⓑWhat will the child’s…
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Algebra and Trigonometry
13.3 Geometric Sequences
Exemplo 1
Finding Common Ratios
Is the sequence geometric? If so, find the common ratio. ⓐ 1,2,4,8,16,... ⓑ 48,12,4, 2,...
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Exemplo 2
Writing the Terms of a Geometric Sequence
List the first four terms of the geometric sequence with a1 =5 and r=–2.
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Exemplo 3
Using Recursive Formulas for Geometric Sequences
Write a recursive formula for the following geometric sequence. {6, 9, 13.5, 20.25,...}
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Exemplo 4
Writing Terms of Geometric Sequences Using the Explicit Formula
Given a geometric sequence with a1 =3 and a4 =24, find a2.
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Exemplo 5
Writing an Explicit Formula for the n th Term of a Geometric Sequence
Write an explicit formula for the nth term of the following geometric sequence. {2, 10, 50, 250,...}
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Exemplo 6
Solving Application Problems with Geometric Sequences
In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year. ⓐWrite a formula for the student population. ⓑEstimate the student population in 2020.
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Algebra and Trigonometry
13.4 Series and Their Notations
Exemplo 1
Using Summation Notation
Evaluate ∑ k=3 7 k^2.
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Exemplo 2
Finding the First n Terms of an Arithmetic Series
Find the sum of each arithmetic series. ⓐ 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 ⓑ 20 + 15 + 10 +…+ −50 ⓒ ∑ k=1 12 3k−8
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Exemplo 3
Solving Application Problems with Arithmetic Series
On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?
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Exemplo 4
Finding the First n Terms of a Geometric Series
Use the formula to find the indicated partial sum of each geometric series. ⓐ S11 for the series 8 + -4 + 2 + … ⓑ ∑^ k=1 6 3⋅2^k
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Exemplo 5
Solving an Application Problem with a Geometric Series
At a new job, an employee’s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.
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Exemplo 6
Determining Whether the Sum of an Infinite Series is Defined
Determine whether the sum of each infinite series is defined. ⓐ 12 + 8 + 4 + … ⓑ 3/4 +1/2 +1/3 +... ⓒ ∑ k=1 ∞ 27⋅ (1/3) ^k ⓓ ∑ k=1 ∞ 5k
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Exemplo 7
Finding the Sum of an Infinite Geometric Series
Find the sum, if it exists, for the following: ⓐ 10+9+8+7+… ⓑ 248.6+99.44+39.776+… ⓒ ∑ k=1 ∞ 4,374⋅ (–1/3) ^(k–1) ⓓ ∑ k=1 ∞ 1/9 ⋅ (4/3) ^k
Exemplo 8
Finding an Equivalent Fraction for a Repeating Decimal
Find an equivalent fraction for the repeating decimal 0.3¯
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Exemplo 9
Solving an Annuity Problem
A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right…
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Algebra and Trigonometry
13.5 Counting Principles
Exemplo 1
Using the Addition Principle
There are 2 vegetarian entrée options and 5 meat entrée options on a dinner menu. What is the total number of entrée options?
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Exemplo 2
Using the Multiplication Principle
Diane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the…
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Exemplo 3
Finding the Number of Permutations Using the Multiplication Principle
At a swimming competition, nine swimmers compete in a race. ⓐHow many ways can they place first, second, and third? ⓑHow many ways can they place first, second, and third if a swimmer named Ariel wins first place?…
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Exemplo 4
Finding the Number of Permutations Using the Formula
A professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and arrange the questions?
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Exemplo 5
Finding the Number of Combinations Using the Formula
A fast food restaurant offers five side dish options. Your meal comes with two side dishes. ⓐHow many ways can you select your side dishes? ⓑHow many ways can you select 3 side dishes?
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Exemplo 6
Finding the Number of Subsets of a Set
A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?
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Exemplo 7
Finding the Number of Permutations of n Non-Distinct Objects
Find the number of rearrangements of the letters in the word DISTINCT.
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Algebra and Trigonometry
13.6 Binomial Theorem
Exemplo 1
Finding Binomial Coefficients
Find each binomial coefficient. ⓐ ( 5 3) ⓑ ( 9 2) ⓒ ( 9 7)
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Exemplo 2
Expanding a Binomial
Write in expanded form. ⓐ (x+y) ^5 ⓑ ( 3x−y) ^4
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Exemplo 3
Writing a Given Term of a Binomial Expansion
Find the tenth term of (x+2y) ^(16) without fully expanding the binomial.
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Algebra and Trigonometry
13.7 Probability
Exemplo 1
Constructing a Probability Model
Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.
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Exemplo 2
Computing the Probability of an Event with Equally Likely Outcomes
A six-sided number cube is rolled. Find the probability of rolling an odd number.
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Exemplo 3
Computing the Probability of the Union of Two Events
A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.
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Exemplo 4
Computing the Probability of the Union of Mutually Exclusive Events
A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.
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Exemplo 5
Using the Complement Rule to Calculate Probabilities
Two six-sided number cubes are rolled. ⓐFind the probability that the sum of the numbers rolled is less than or equal to 3. ⓑFind the probability that the sum of the numbers rolled is greater than 3.
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Exemplo 6
Computing Probability Using Counting Theory
A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears. ⓐFind the probability that only bears are chosen. ⓑFind the probability that 2 bears and 3 dogs are chosen. ⓒFind the probability…
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