Consecutive integers summing

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. 

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 ... 

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. 

The formula for finding the sum of a list of consecutive integers requires that you have the first and last terms in the list. The multiples of 4 are all four units apart. You have an arithmetic sequence with terms the difference of which is 4. (You can find more on arithmetic sequences in Setting the stage for the sums, earlier in this chapter.) So, to find the 20th term in the list of multiples of 4 that start with 60, use the formula an = + d(n - 1), giving you... 

Adding up lists of numbers is always a huge amount of fun — or not. It depends on what you like to do with your leisure time, I suppose. When formulas are available to make arithmetic computations easier and more accurate, you jump at the chance to use those formulas. Here I give you some applications of sums of consecutive integers. 

Find the sum of the consecutive integers 1, 2, 3, 4, 5,... 19, 20 by using the formula for the sum of consecutive integers. The first term is 1 the last term is 20 and the number of terms, n, is 20. Compare that sum with the number of blocks to see if Jimmy will have enough blocks. 

Magic square is an unusual numerical configuration containing consecutive integers in arrangements so that the sum of numbers in any row, column, or diagonal are identical. Such squares were known approximately 4,000 years ago in China. 

The sum of 3 consecutive integers is 3 7 more than the largest integer. ... 

This is a linear function of previous values of Y and so can be written in form of Equation (5.40). To determine the coefficients B) we use the formula for the sum of a series of consecutive integers ... 

We have used the fact that the sum of a set of consecutive integers is the mean of the first integer and the last integer times the number of members of the set (a fact reportedly first discovered by Gauss when he was seven years old). 

Which of the following lists three consecutive even integers whose sum is 30 ... 

The four sons in the Johnson family have ages that are consecutive even integers. If the sum of their ages is 84, how old is the youngest ... 

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