If you know exactly how to solve word difficulties involving the sum of consecutive even integers, you should have the ability to easily resolve word difficulties that indicate the sum of continuous odd integers. The vital is to have a good grasp of what weird integers are and also how continually odd integers can be represented.


Odd Integers

If friend recall, an even integer is always 2 times a number. Thus, the general type of an also number is n=2k, whereby k is an integer.

So what go it typical when us say the an integer is odd? Well, it means that it’s one less or one more than an even number. In other words, odd integers space one unit much less or one unit more of an also number.

Therefore, the general type of one odd integer deserve to be expressed as n is n=2k-1 or n=2k+1, where k is an integer.


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Observe that if you’re offered an also integer, that also integer is always in in between two weird integers. Because that instance, the also integer 4 is between 3 and also 5.

To illustrate this an easy fact, take a look in ~ the chart below.

You are watching: Sum of 3 consecutive odd integers


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As you deserve to see, no matter what also integer us have, that will always be in in between two strange integers. This diagram also illustrates the an strange integer have the right to be stood for with one of two people n=2k-1 or n=2k+1, where k is an integer.

Consecutive weird Integers

Consecutive odd integers space odd integers the follow each other in sequence. Friend may uncover it difficult to believe, but as with even integers, a pair of any kind of consecutive strange integers are also 2 devices apart. Simply put, if girlfriend select any odd integer from a collection of consecutive odd integers, climate subtract it by the previous one, their distinction will be +2 or just 2.

Here are some examples:


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When addressing word problems, it really doesn’t matter which general creates of an odd integer friend use. Even if it is you use 2k-1 or 2k+1, the final solution will be the same.

To prove it come you, us will resolve the an initial word difficulty in 2 ways. Then because that the rest of the word problems, we will certainly either usage the kind 2k-1 or 2k+1.

instances of fixing the amount of continuous Odd Integers

Example 1: find the three consecutive strange integers whose amount is 45.

METHOD 1

We will fix this word trouble using 2k+1 which is among the general creates of one odd integer.

Let 2k+1 be the first weird integer. Because odd integers are also 2 devices apart, the second consecutive odd integer will be 2 much more than the first. Therefore, \left( 2k + 1 \right) + \left( 2 \right) = 2k + 3 whereby 2k + 3 is the second continually odd integer. The third strange integer will certainly then it is in \left( 2k + 3 \right) + \left( 2 \right) = 2k + 5.

The amount of our 3 consecutive weird integers is 45, for this reason our equation setup will be:


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Now that we have actually our equation, let’s proceed and also solve because that k.


At this point, we have actually the worth for k. However, note that k is not the first odd integer. If you review the equation above, the an initial consecutive odd creature is 2k+1. Therefore instead, we will use the value of k in bespeak to find the an initial consecutive weird integer. Therefore,


We’ll usage the value of k again to determine what the 2nd and 3rd odd integers are.

Second weird integer:


Third odd integer:


Finally, let’s inspect if the amount of the three consecutive strange integers is indeed 45.


Final price (Method 1): The three consecutive odd integers space 13, 15, and also 17, which when added, results to 45.


METHOD 2

This time, us will resolve the word trouble using 2k-1 i beg your pardon is additionally one of the general creates of one odd integer.

Let 2k-1 it is in the first continually odd integer. As discussed in an approach 1, weird integers are additionally 2 devices apart. Thus, we have the right to represent our second continually odd integer as \left( 2k - 1 \right) + \left( 2 \right) = 2k + 1 and also the third consecutive odd integer together \left( 2k + 1 \right) + \left( 2 \right) = 2k + 3.

1st weird integer: 2k-1 second odd integer: 2k+13rd weird integer: 2k+3

Now that we know how to represent each consecutive odd integer, we simply have to translate “three continually odd integers whose amount is 45” right into an equation.


Proceed and also solve because that k.


Let’s currently use the worth of k i m sorry is k=7, to determine the three consecutive integers

First weird integer:
Second odd integer:
Third weird integer:

The last action for us to carry out is come verify the the sum of 13, 15, and also 17 is in fact, 45.


Final answer (Method 2): The three consecutive odd integers whose sum is 45 room 13, 15, and also 17.

PROBLEM WRAP-UP: therefore what have we learned while fixing this difficulty using 2k-1 and also 2k+1? Well, come start, we were may be to view that even if it is we provided 2k-1 or 2k+1, we still obtained the same 3 consecutive weird integers 13, 15, and also 17 whose sum is 45, as such satisying the given facts in our initial problem. So, that is clear that it doesn’t matter what general form of strange integers us use. Whether it’s 2k-1 or 2k+1, we will still come at the same last solution or answer.

Example 2: The amount of four consecutive strange integers is 160. Find the integers.

Before us start resolving this problem, let’s identify the important facts the are provided to us.

What perform we know?

The integers room odd and are consecutiveThe amount of the continuous integers is 160 which likewise implies the we need to add the integersThe integers different by 2 unitsEach creature is 2 more than the vault integer

With this facts in mind, we deserve to now represent our 4 consecutive strange integers. But although we deserve to use either of the 2 general creates of strange integers, i.e. 2k-1 or 2k+1, we’ll only use 2k+1 to stand for our first odd continually integer in this problem.

Let 2k+1, 2k+3, 2k+5 , and also 2k+7 be the 4 consecutive strange integers.


Proceed by creating the equation then resolve for k.


Alright, for this reason we obtained k=18. Is this our first odd integer? The answer is, no. Again, remember the k is no the first odd integer. Yet instead, we’ll usage its worth to find what our consecutive odd integers are.


What’s left for us to execute is to inspect if 160 is undoubtedly the amount of the consecutive odd integers 37, 39, 41, and also 43.


Example 3: discover the three consecutive weird integers whose sum is -321.

Important Facts:

We require to include three integers that are consecutiveSince the integers space odd, they are 2 devices apartThe sum of the three consecutive odd integers have to be -321 The sequence of strange integers will more likely involve negative integers

Represent the three consecutive odd integers. Because that this problem, us will use the general type 2k-1 to represent our first consecutive odd integer. And since odd integers room 2 units apart, then we have actually 2k+1 as our second, and 2k+3 together our 3rd consecutive integer.


Next, analyze “three consecutive odd integers whose sum is -321” right into an equation and also solve because that k.

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Take the worth of k i m sorry is -54 and also use it to determine the 3 consecutive strange integers.


Finally, verify that as soon as the 3 consecutive weird integers -109, -107 ,and -105 room added, the amount is -321.