If you know how to settle word difficulties involving the amount of consecutive even integers, friend should have the ability to easily resolve word problems that indicate the sum of consecutive odd integers. The an essential is to have a good grasp that what strange integers are and how continuous odd integers have the right to be represented.
If friend recall, an even integer is constantly 2 time a number. Thus, the general form of an even number is n=2k, wherein k is one integer.
So what go it average when us say that an essence is odd? Well, it way that it’s one less or one more than an also number. In other words, odd integers are one unit much less or one unit much more of an also number.
Therefore, the general type of an odd integer can be expressed as n is n=2k-1 or n=2k+1, whereby k is an integer.
Observe that if you’re given an even integer, that also integer is always in in between two weird integers. Because that instance, the also integer 4 is in between 3 and also 5.
To highlight this basic fact, take it a look in ~ the diagram below.
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As you have the right to see, no issue what also integer we have, the will constantly be in in between two weird integers. This diagram additionally illustrates that an weird integer can be stood for with either n=2k-1 or n=2k+1, whereby k is one integer.
Consecutive strange Integers
Consecutive strange integers are odd integers the follow each other in sequence. Friend may find it difficult to believe, but as with even integers, a pair of any consecutive odd integers are likewise 2 devices apart. Simply put, if friend select any kind of odd integer native a set of consecutive odd integers, climate subtract it by the ahead one, their difference will be +2 or just 2.
Here space some examples:
When solving word problems, it yes, really doesn’t matter which general forms of an odd integer girlfriend use. Whether you use 2k-1 or 2k+1, the final solution will certainly be the same.
To prove it come you, us will fix the very first word difficulty in 2 ways. Then for the remainder of the word problems, we will either usage the kind 2k-1 or 2k+1.
instances of fixing the amount of continually Odd Integers
Example 1: find the 3 consecutive weird integers whose sum is 45.
We will settle this word difficulty using 2k+1 which is among the general creates of one odd integer.
Let 2k+1 it is in the first odd integer. Because odd integers are also 2 devices apart, the second consecutive strange integer will be 2 more than the first. Therefore, left( 2k + 1 ight) + left( 2 ight) = 2k + 3 wherein 2k + 3 is the second continuous odd integer. The third weird integer will then be left( 2k + 3 ight) + left( 2 ight) = 2k + 5.
The amount of our 3 consecutive odd integers is 45, so our equation setup will certainly be:
Now that we have our equation, let’s proceed and solve for k.
At this point, we have the value for k. However, keep in mind that k is no the first odd integer. If you evaluation the equation above, the first consecutive odd integer is 2k+1. So instead, we will usage the value of k in stimulate to find the an initial consecutive weird integer. Therefore,
We’ll use the worth of k again to determine what the 2nd and 3rd odd integers are.
Second strange integer:
Third strange integer:
Finally, let’s inspect if the amount of the three consecutive odd integers is indeed 45.
Final answer (Method 1): The 3 consecutive strange integers space 13, 15, and 17, which when added, results to 45.
This time, us will solve the word problem using 2k-1 i m sorry is additionally one of the general develops of one odd integer.
Let 2k-1 be the first continually odd integer. As debated in an approach 1, strange integers are also 2 units apart. Thus, we can represent our second continually odd integer as left( 2k - 1 ight) + left( 2 ight) = 2k + 1 and also the third consecutive odd integer together left( 2k + 1 ight) + left( 2 ight) = 2k + 3.1st weird integer: 2k-1 2nd odd integer: 2k+13rd odd integer: 2k+3
Now that us know just how to represent each continually odd integer, we simply need to translate “three continuous odd integers whose sum is 45” into an equation.
Proceed and also solve for k.
Let’s now use the value of k i m sorry is k=7, to identify the 3 consecutive integersFirst odd integer:
Second weird integer:
Third strange integer:
The last action for us to execute is come verify that the sum of 13, 15, and 17 is in fact, 45.
Final prize (Method 2): The 3 consecutive odd integers whose sum is 45 space 13, 15, and also 17.
PROBLEM WRAP-UP: so what have we learned while resolving this difficulty using 2k-1 and 2k+1? Well, come start, us were able to watch that even if it is we provided 2k-1 or 2k+1, us still gained the same 3 consecutive strange integers 13, 15, and also 17 whose amount is 45, because of this satisying the provided facts in our original problem. So, it is clear the it doesn’t matter what general form of strange integers us use. Even if it is it’s 2k-1 or 2k+1, we will certainly still arrive at the same final solution or answer.
Example 2: The sum of four consecutive weird integers is 160. Uncover the integers.
Before us start solving this problem, let’s identify the essential facts the are provided to us.
What execute we know?The integers room odd and are consecutiveThe amount of the continually integers is 160 which additionally implies the we require to include the integersThe integers different by 2 unitsEach integer is 2 much more than the previous integer
With this facts in mind, we can now stand for our 4 consecutive weird integers. But although we have the right to use one of two people of the two general forms of weird integers, i.e. 2k-1 or 2k+1, we’ll only use 2k+1 to represent our first odd continually integer in this problem.
Let 2k+1, 2k+3, 2k+5 , and 2k+7 it is in the 4 consecutive odd integers.
Proceed by writing the equation then fix for k.
Alright, so we obtained k=18. Is this our an initial odd integer? The price is, no. Again, remember the k is no the first odd integer. But instead, we’ll usage its value to find what ours consecutive odd integers are.
What’s left for us to carry out is to check if 160 is certainly the sum of the consecutive odd integers 37, 39, 41, and 43.
Example 3: uncover the 3 consecutive strange integers whose amount is -321.
Important Facts:We need to add three integers that space consecutiveSince the integers room odd, they space 2 systems apartThe sum of the 3 consecutive weird integers should be -321 The sequence of weird integers will more likely involve negative integers
Represent the three consecutive strange integers. Because that this problem, us will use the general form 2k-1 to represent our very first consecutive strange integer. And also since weird integers space 2 devices apart, climate we have 2k+1 as our second, and also 2k+3 together our 3rd consecutive integer.
Next, interpret “three continually odd integers whose amount is -321” into an equation and solve for k.
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Take the worth of k which is -54 and use the to determine the three consecutive strange integers.
Finally, verify that when the three consecutive strange integers -109, -107 ,and -105 room added, the amount is -321.