Integers consecutive

These methods may be called analytical, by contrast with another class of iterations that might be called arithmetic, since they exploit the fact that the number representation is finite and digital. The familiar Homer s method is an example. The first step is to establish that a root lies between a certain pair of consecutive integers. Next, if the representation is decimal, f(x) is evaluated at consecutive tenths to determine the pair of consecutive tenths between which the root lies. This is repeated for the hundredths, thousandths, etc., to as many places as may be desired and justified. 

Throughout the course of this book, we have looked at many word problems. Several problems involving distance and speed, percents, simple interest, and ratio and proportions have been reviewed. One other type of word problem not reviewed previously is consecutive integer problems. These problems are relatively easy to solve on multiple-choice tests. 

CONSECUTIVE INTEGERS integers that follow one after the other in order. They differ by one. Example 4, 5, 6, 7, 8, 9. ... ... 

The guess and check strategy is an effective way to solve consecutive integer problems. 

Carlos s and James s ages are consecutive integers. James is older. Four years ago, Carlos was half the age that James is now. How old is James now ... 

Number problems can contain situations involving one, two, or even more different integers, whole numbers, or fractions. The more numbers you have to solve for, the more interesting the problem becomes. Usually, when more than one number is involved in one of these problems, there s some sort of relationship between the numbers — some mathematical comparison. Chapter 12 is completely devoted to consecutive integers, so you ll find other types of problems here. Other problems requiring that you find two or more solutions are solved with systems of equations. (You ll find systems in Chapter 17.)... 

Creating lists of consecutive integers from descriptions Focusing on consecutive odds and evens Solving for one of several in a list Applying consecutive integers to practical problems... 

In this chapter, you find consecutive integers, consecutive even integers, consecutive multiples of fives, and so on. The word problems come in as puzzles to find the first, the middle, or the last in a list of consecutive integers. After an introduction on ways to find the sum of a large number of consecutive numbers, you ll see some interesting applications from seating charts to orchards. 

Any list of consecutive integers can be described in many different ways. You may give the overall pattern that describes how far apart the numbers in the list are, and then you give one of the numbers in the list and its position. Or... 

When writing a list of consecutive integers, you let the first integer in the list be signified with a variable. Some people like to use x, because that s the universal unknown or variable. In this chapter, you see a consistent use of another variable, n. Using n for consecutive integers is pretty standard notation, too. 

Following are more examples of the algebraic representation of lists of consecutive integers. You choose the number for n, and the rest fall in line. Of course, if you want even integers, you have to pick an even integer for n ... 

When solving consecutive integer problems, you solve for n and use that value to answer some question about the list or some number on that list. 

If n = 4 and you re trying to find three consecutive integers, then your three integers are 4, 5, and 6. 

Consecutive integer problems have a common theme Take a list of consecutive integers, perform a certain operation, and you get a numerical result. When writing the equations needed to solve these problems, you use a fairly common pattern Let n represent the first number in the list, let n + d (where d is the common difference) be the second number, let n + 2d be the third number, and so on. The most common operation performed in these problems is addition — so add em up. 

The Problem The sum of three consecutive integers is 45. What are the integers ... 

The Problem The sum of four consecutive integers is 38. What is the largest number ... 

The Problem The sum of eight consecutive integers is 4. What is their product ... 

You may think that there s been some error here. How can eight consecutive integers add up to a number smaller than the number of integers The answer to that question is Negative numbers. Keep that in mind as you answer the question posed in this problem. 

Consecutive integers are lists of integers that have a common difference between the terms. Odd and even numbers have a common difference of two between each term. Multiples of three have a common difference of three between the consecutive terms, and so on. 

Doing more than addition to consecutive integers... 

Write the two consecutive integers as n and n + 1. Multiply them together and set the equation equal to 20. You get a quadratic equation that s solved by setting the equation equal to 0, factoring, and solving for the numbers. 

The quadratic equation gives you two different answers. When n = 4, you get the two consecutive integers 4 and 5. The product of 4 and 5 is, indeed, 20. But what about the solution n = -5 If n = -5, then n+l=-5+l = -4. The product of -5 and -4 is also 20. This problem has two different solutions. As long as you re happy with negative integers, too, then you accept both sets of answers. 

The solution n = 11 is used to find the other odd integer by adding 2. The two consecutive integers are 11 and 13. You don t bother with the solution n = -13, because the problem specifies that you re to find positive integers. 

The numbers 1, 2, 3, and 4 are four consecutive integers. The numbers 1, 4, 9, and 16 are four consecutive squares of those first four numbers. And the cubes of the same four numbers are 1, 8, 27, and 64. Squares and cubes of consecutive numbers are squared and cubed, respectively, before being added together or having some other operation performed upon them. 

A sequence of numbers is a list of numbers created by a particular pattern or mathematical rule. An arithmetic sequence is a list of numbers in which there is a common difference between the consecutive numbers in the sequence. So consecutive integers are a special type of arithmetic sequence. The rule that allows you to add up any number of terms in an arithmetic sequence also lets you solve some problems involving the sums of consecutive integers. 

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