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Chapter 9: Problem 4

Fill in the blanks If the ________ differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model.

Short Answer

Expert verified
'second differences'

Step by step solution

01

Understanding the statement

Let's understand the given statement. It explains a property of sequences, particularly a perfect quadratic sequence. The property is that if a particular kind of differences in a sequence is the same nonzero number, then the sequence can be modeled using a perfect quadratic equation. The term that we need to fill in the blank is related to the type of differences in the sequence that need to be identical for the sequence to be a perfect quadratic sequence.
02

Identifying the correct term

The differences that need to be the same for a sequence to be a perfect quadratic sequence are 'second differences'. This is because in a quadratic sequence, the first differences (the differences between consecutive terms) may not be the same, but the second differences (the differences of the first differences) are constant.
03

Filling in the blank

Since we have identified that the correct term is 'second differences', we can now fill in the blank in the original sentence: 'If the second differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model.'

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Second Differences
In a sequence, when we talk about differences, we primarily focus on two types: first differences and second differences. Second differences are particularly important when analyzing sequences to determine if they can be modeled by a quadratic equation.

Here's how it works:
  • First, you calculate the first differences by subtracting consecutive terms in the sequence.
  • Next, you take those first difference results and find their differences; these are called second differences.
For a sequence to fit a quadratic model, its second differences must be constant (and not zero). This consistency of second differences is a hallmark of quadratic behavior in sequences.

If the second differences are the same number throughout, you can be confident that the sequence can be modeled as a quadratic equation, indicating that it's not just a simple arithmetic or linear sequence.
Sequences
Sequences are ordered lists of numbers that follow a particular pattern or rule. Understanding the type of sequence can help in modeling it accurately. There are various types of sequences, but when examining for quadratic properties, recognizing the pattern is crucial.

Here are some basic characteristics of sequences:
  • Each number in the sequence is called a term.
  • The difference between terms, either first or second, can reveal much about the sequence type.
  • If first differences are constant, it suggests a linear relationship.
  • If second differences are constant, it suggests a quadratic relationship.
Recognizing sequences is beneficial in mathematics because it allows for predicting future terms, recognizing equations that describe real-world phenomena, and aiding in various fields such as finance, computer science, and physics.
Quadratic Equations
Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are crucial as they can model many real-world scenarios, such as projectile motion or optimization problems.

Some important aspects of quadratic equations include:
  • The solutions are called roots, which can be real or complex.
  • Graphically, they form a parabola, which can open upwards or downwards depending on the sign of \( a \).
  • The vertex of the parabola represents the maximum or minimum value of the function.
When you discover that the second differences of a sequence are constant, you can confidently proceed to model it using a quadratic equation. This adaptability makes understanding quadratic equations valuable across various mathematical and practical applications.

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Most popular questions from this chapter

In Exercises 1 - 7, fill in the blanks. If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________.

Consider a group of people. (a) Explain why the following pattern gives the probabilities that the people have distinct birthdays. \( n = 2: \dfrac{365}{365} \cdot \dfrac{364}{365} = \dfrac{365 \cdot 364}{365^2} \) \( n = 3: \dfrac{365}{365} \cdot \dfrac{364}{365} \cdot \dfrac{363}{365} = \dfrac{365 \cdot 364 \cdot 363}{365^3} \) (b) Use the pattern in part (a) to write an expression for the probability that \( n = 4 \) people have distinct birthdays. (c) Let \( P_n \) be the probability that the \( n \) people have distinct birthdays. Verify that this probability can be obtained recursively by \( P_1 = 1 \) and \( P_n = \dfrac{365 - (n - 1)}{365} P_{n - 1} \). (d) Explain why \( Q_n = 1 - P_n \) gives the probability that at least two people in a group of \( n \) people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than \( \dfrac{1}{2} \)? Explain.

In Exercises 71 and 72, determine whether the statement is true or false. Justify your answer. Rolling a number less than \( 3 \) on a normal six-sided die has a probability of \( \dfrac{1}{3} \) . The complement of this event is to roll a number greater than \( 3 \), and its probability is \( \dfrac{1}{2} \).

In Exercises 15 - 20, find the probability for the experiment of tossing a coin three times. Use the sample space \( S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \). The probability of getting at least one head

A college sends a survey to selected members of the class of 2009. Of the 1254 people who graduated that year, 672 are women, of whom 124 went on to graduate school. Of the 582 male graduates, 198 went on to graduate school. An alumni member is selected at random. What are the probabilities that the person is (a) female, (b) male, and (c) female and did not attend graduate school?
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