Ch. 4_Consecutive Integers Strategy

General facts to note about sums and averages of evenly spaced sets (especially sets of consecutive integers). 2

-On the other hand, the average of an even number of consecutive integers (1, 2, 3, 4) will never be an integer (2, 5), because there is no true "middle number"

Consecutive integers

-Special cases of consecutive mutliples: all of the values in the set increase by 1, and all integers are multiples of 1. -Increments of 1. E.g., 12, 13, 14, 15, 16

Consecutive multiples

-Special cases of evenly spaced sets: all of the values in the set are multiples of the increment. -Multiples of the increment. E.g., 12, 16, 20, 24 Increase from one to the next by 4, and each element is a multiple of 4.

To find the sum of consecutive integers Example Sum of integers 10 to 50 would be

(10 + 50)/2 x (50 - 10 + 1) = 1,230

Counting integers For consecutive integers formula

(Last - First + 1)

Counting integers For consecutive multiples formula

(Last - First) / Increment + 1 *The bigger the increment, the smaller the result, because there is a larger gap between the numbers you are counting. *Sometimes it's easier to list the terms of a consecutive pattern and count them, especially if the list is short or if one or both of the extremes are omitted.

The relations among evenly spaced sets, consecutive multiples, and consecutive integers

-All sets of consecutive integers are sets of consecutive multiples. -All sets of consecutive multiples are evenly spaced sets. -All evenly spaced sets are fully defined if the following 3 parameters are known: 1. The smallest (first) or largest (last) number in the set. 2. The increment (always 1 for consecutive integers) 3. The number of items in the set.

General facts to note about sums and averages of evenly spaced sets (especially sets of consecutive integers). 1

-Average of an odd number of consecutive integers (1, 2, 3, 4, 5) will always be an integer (3). This is because the "middle number" will be a single integer.

Evenly spaced sets

-These are sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next. -Constant increments. E.g., 4, 7, 10, 13, 16 Value increases by 3 over the previous value.

Find the sum of any 4 consecutive integers

1 + 2 + 3 + 4 = 10 8 + 9 + 10+ 11 = 38 Notice that NEITHER sum is a multiple of 4. In other words, both sums are not divisible by 4.

2. The mean and median of the set are equal to the average of the FIRST and LAST terms. Example What is the arithmetic mean of 4, 8, 12, 16, and 20?

20 is the largest (last) number and 4 is the smallest (first). The mean and median are therefore equal to (20+4) / 2 =12.

2. The mean and median of the set are equal to the average of the FIRST and LAST terms. Example What is the arithmetic mean of 4, 8, 12, 16, 20, and 24?

24 is the largest (last) number and 4 is the smallest (first). The mean and median are therefore equal to (24+4) / 2=14.

If the sum of the last 3 integers in a set of 7 consecutive integers is 258, what is the sum of the first 4 integers?

330. Think of the set as n, (n+1), (n+2), (n+3), (n+4), (n+5), (n+6). (n+4), (n+5), (n+6) = 3n + 15 = 258. Therefore, n = 81. The sum of the first 4 integers is 81 + 82 + 83 + 84 = 330. Or, the sum of the first four integers is 4n + 6. If n = 81, then 4(81) + 6 = 330. Was on the right track, but got wrong. I got 310. I counted backwards 4 times (76 + 77 + 78 + 79).

Sums of consecutive integers and divisibility Find the sum of any 5 consecutive integers

4 + 5 + 6 + 7 + 8 = 30. 13 + 14 + 15 + 16 + 17 = 75 Notice that both sums are multiples of 5. In other words, both sums are divisible by 5.

What is the sum of all the positive integers up to 100, inclusive?

5,050. There are 100 integers. The number exactly in the middle in 50.5. Therefore, multiply 100 by 50.5 to find the sum of all the integers in the set. Guessed.

Counting integers Example How many integers are there from 14 to 765, inclusive?

765-14 + 1 = 752

Counting integers

Add one before you are done.

To find the sum of consecutive integers

Average of numbers x Number of terms

Any two even numbers in a multiplication will

Ensure the product be divisible by 4.

Consecutive integers

Integers that follow one after another from a given starting point, without skipping any integer. E.g., 4, 5, 6, and 7

Consecutive integers and divisibility

Put prime boxes next to each other to show the factors of consecutive integers.

Products of consecutive integers and divisibility

The result is that the product of any set of 3 consecutive integers is divisible by 3. This rule applies to any number of consecutive integers:

Products of consecutive integers and divisibility This rule applies to any number of consecutive integers:

The product of k consecutive integers is always divisible by k factorial (k!).

1. Arithmetic mean (average) and median are equal to each other.

The average of the elements in the set can be found by figuring out the median, or middle number.

Properties of evenly spaced sets

The following properties apply to ALL evenly spaced sets: 1. Arithmetic mean (average) and median are equal to each other. 2. The mean and median of the set are equal to the average of the FIRST and LAST terms. 3. The sum of the elements in the set equals the arithmetic mean number in the set times the number of times in the set. *For evenly spaced sets, the median equals the mean, and it also equals the average of the first and last numbers in the set.

For any set of consecutive integers with an ODD number of items

The sum of all the integers is ALWAYS a multiple of the number of items. This is because the sum equals the average times the number of times. For an odd number of integers, the average is an integer, so the sum is a multiple of the number of items.

For any set of consecutive integers with an EVEN number of items,

The sum of all the items is NEVER a multiple of the number of items. This is because the sum equals the average times the number of items.

1. Arithmetic mean (average) and median are equal to each other. Example What is the arithmetic mean of 4, 8, 12, 16, and 20?

We have five consecutive multiples of four. The median is the 3rd largest, or 12. Since this is an evenly spaced set, the mean is also 12.

3. The sum of the elements in the set equals the arithmetic mean number in the set times the number of times in the set. Example What is the arithmetic mean of 4, 8, 12, 16, and 20?

it is equal to 12. There are 5 terms, so the sum equals 12 x 5 = 60.

3. The sum of the elements in the set equals the arithmetic mean number in the set times the number of times in the set. Example What is the arithmetic mean of 4, 8, 12, 16, 20, and 24?

it is equal to 14. There are 6 terms, so the sum equals 14 x 6=84.

*For all evenly, just remember

the average equals (first+last)/2.

General facts to note about sums and averages of evenly spaced sets (especially sets of consecutive integers). 3

This is because consecutive integers alternate between EVEN and ODD numbers. Therefore, the "middle number" for an even number of consecutive integers is the average of two consecutive integers, which is never an integer.

Other types of consecutive patterns

-Consecutive even integers (8, 10, 12, 14) -Consecutive primes (11, 13, 17, 19)

Products of consecutive integers and divisibility Example

1 x 2 x 3 = 6 2 x 3 x 4 = 24 6 x 7 x 8 = 336

The same logic applies to a set of 4 consecutive integers, 5 consecutive integers, and other number of consecutive integers.

For instance, the product of any set of 4 consecutive integers will be divisible by 4...since that set will always contain one multiple of 4, at least one multiple of 3, and another even number (a multiple of 2).

Will the average of 6 consecutive integers be an integer?

I said yes. No. For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER a multiple of the number of items. For example, 4, 5, 6, 7, 8, and 9. 39/6 = 6.5

Products of consecutive integers and divisibility According to the Factor Foundation Rule, every number is divisible by all the factors of its factors.

If there is always a multiple of 3 in a set of 3 consecutive integers, the product of 3 consecutive integers will always be divisible by 3. Additionally, there will always be at least one multiple of 2 (an even number) in any set of 3 consecutive integers. Therefore, the product of 3 consecutive integers will also be divisible by 2.

Counting integers For consecutive multiples formula Example How many multiples of 7 are there between 100 and 150?

It may be easiest to list the multiples: 105, 112, 119, 126, 133, 140, 147. Count the number of terms to get the answer: 7. Alternatively, we could note that 105 is the first number, 147 is the last number, and 7 is the increment. Number of terms = (Last - First) / Increment + 1 = (147-105)/7 +1 = 6 + 1 = 7.

Is the sum of the integers from 54 to 153, inclusive, divisible by 100?

No. I said yes. I calculated the sum, which was 10,350, which is divisible by 100. However, this violates the rule. There are 100 integers from 54 to 153, inclusive. For any even number of consecutive integers, the sum of all the integers is NEVER a multiple of the numbers of integers.

3. The sum of the elements in the set equals the arithmetic mean number in the set times the number of times in the set.

This property applies to all sets, but it takes on special significance in the case of evenly spaced sets because the average is not only the mean, but also the median.

The sum of consecutive integers Example What is the sum of all integers from 20 to 100, inclusive?

Using the rules of evenly spaced sets, we can use shortcuts: 1. Average the first and last term to find the precise "middle" of the set. 2. Count the number of terms: 100-20=80, plus 1 yields 81. 3. Multiply the "middle" number by the number of terms to find the sum: 60x81=4,860.

In a sequence of 8 consecutive integers, how much greater is the sum of the last four integers than the sum of the first four integers?

Was unsure, but got right. Careless miscalculation. 16. Think of the set of 8 consecutive integers as follows: n, (n+1), (n+2), (n+3), (n+4), (n+5), (n+6), (n+7) This becomes (4n+22) - (4n+6) = 22 - 6 = 16

1. Arithmetic mean (average) and median are equal to each other. Example What is the arithmetic mean of 4, 8, 12, 16, 20, and 24?

We have six consecutive multiples of four. The median is the mean of the 3rd largest and 4th largest, or the average of 12 and 16. The median is 14. Since this is an evenly spaced set, the average if also 14.

Product of two even numbers is always

divisible by 4