If girlfriend know just how to fix word troubles involving the sum of consecutive also integers, girlfriend should have the ability to easily resolve word difficulties that indicate the sum of continuous odd integers. The crucial is to have a good grasp of what odd integers are and also how consecutive odd integers have the right to be represented.
If you recall, an even integer is constantly 2 times a number. Thus, the general kind of an even number is n=2k, wherein k is one integer.
So what walk it typical when us say the an integer is odd? Well, it method that it’s one less or one more than an also number. In various other words, odd integers are one unit much less or one unit more of an also number.
Therefore, the general kind of one odd integer have the right to be expressed as n is n=2k-1 or n=2k+1, wherein k is one integer.
Observe the if you’re provided an even integer, that also integer is constantly in in between two weird integers. For instance, the even integer 4 is between 3 and also 5.
To illustrate this basic fact, take a look in ~ the chart below.
You are watching: The sum of two consecutive odd integers
As you deserve to see, no matter what also integer us have, that will always be in in between two odd integers. This diagram also illustrates that an odd integer can be stood for with one of two people n=2k-1 or n=2k+1, wherein k is one integer.
Consecutive strange Integers
Consecutive strange integers are odd integers the follow each various other in sequence. You may discover it hard to believe, but as with even integers, a pair of any type of consecutive strange integers are additionally 2 systems apart. Simply put, if friend select any type of odd integer native a set of continually odd integers, climate subtract the by the vault one, their distinction will be +2 or simply 2.
Here room some examples:
When addressing word problems, it yes, really doesn’t issue which general forms of an odd integer girlfriend use. Whether you use 2k-1 or 2k+1, the final solution will be the same.
To prove it come you, us will fix the first word trouble in two ways. Then because that the remainder of the word problems, we will either usage the form 2k-1 or 2k+1.
examples of addressing the sum of consecutive Odd Integers
Example 1: uncover the three consecutive odd integers whose sum is 45.
We will settle this word difficulty using 2k+1 which is among the general develops of an odd integer.
Let 2k+1 it is in the first odd integer. Since odd integers are additionally 2 units apart, the 2nd consecutive weird integer will certainly be 2 much more than the first. Therefore, \left( 2k + 1 \right) + \left( 2 \right) = 2k + 3 where 2k + 3 is the second continually odd integer. The third odd integer will certainly then be \left( 2k + 3 \right) + \left( 2 \right) = 2k + 5.
The amount of our 3 consecutive odd integers is 45, so our equation setup will be:
Now the we have our equation, let’s proceed and also solve for k.
At this point, we have actually the value for k. However, note that k is not the an initial odd integer. If you testimonial the equation above, the an initial consecutive odd integer is 2k+1. For this reason instead, we will usage the value of k in bespeak to discover the very first consecutive weird integer. Therefore,
We’ll usage the worth of k again to identify what the second and 3rd odd integers are.
Second strange integer:
Third weird integer:
Finally, let’s examine if the amount of the 3 consecutive weird integers is indeed 45.
Final price (Method 1): The 3 consecutive strange integers room 13, 15, and 17, which once added, results to 45.
This time, we will settle the word difficulty using 2k-1 i m sorry is additionally one the the general creates of an odd integer.
Let 2k-1 be the first continuous odd integer. As discussed in method 1, odd integers are additionally 2 devices apart. Thus, we have the right to represent ours second consecutive odd integer together \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 strange integer: 2k-1 2nd odd integer: 2k+13rd odd integer: 2k+3
Now that we know just how to stand for each continually odd integer, us simply need to translate “three consecutive odd integers whose amount is 45” into an equation.
Proceed and solve for k.
Let’s currently use the value of k i beg your pardon is k=7, to recognize the 3 consecutive integersFirst weird integer:
Second strange integer:
Third strange integer:
The last action for us to carry out is to verify that the amount of 13, 15, and also 17 is in fact, 45.
Final price (Method 2): The three consecutive odd integers whose sum is 45 room 13, 15, and 17.
PROBLEM WRAP-UP: for this reason what have we learned while addressing this trouble using 2k-1 and 2k+1? Well, come start, we were maybe to watch that even if it is we supplied 2k-1 or 2k+1, we still obtained the same 3 consecutive strange integers 13, 15, and 17 whose sum is 45, as such satisying the offered facts in our original problem. So, that is clear the it doesn’t issue what general type of odd integers we use. Whether it’s 2k-1 or 2k+1, we will certainly still come at the same last solution or answer.
Example 2: The sum of 4 consecutive odd integers is 160. Discover the integers.
Before we start fixing this problem, let’s identify the vital facts the are offered to us.
What do we know?The integers room odd and are consecutiveThe sum of the continuous integers is 160 which also implies the we need to include the integersThe integers differ by 2 unitsEach essence is 2 much more than the ahead integer
With these facts in mind, we deserve to now stand for our 4 consecutive odd integers. Yet although we have the right to use either of the two general develops of odd integers, i.e. 2k-1 or 2k+1, we’ll just use 2k+1 to stand for our first odd consecutive integer in this problem.
Let 2k+1, 2k+3, 2k+5 , and 2k+7 it is in the four consecutive weird integers.
Proceed by composing the equation then resolve for k.
Alright, therefore we obtained k=18. Is this our an initial odd integer? The price is, no. Again, remember that k is no the very first odd integer. Yet instead, we’ll usage its value to uncover what our consecutive strange integers are.
What’s left for us to execute is to examine if 160 is indeed the sum of the continually odd integers 37, 39, 41, and also 43.
Example 3: discover the three consecutive odd integers whose amount is -321.
Important Facts:We need to add three integers that room consecutiveSince the integers are odd, they room 2 systems apartThe amount of the three consecutive strange integers must be -321 The succession of strange integers will more likely involve an unfavorable integers
Represent the three consecutive odd integers. Because that this problem, us will usage the general form 2k-1 to stand for our very first consecutive odd integer. And also since odd integers space 2 units apart, then we have actually 2k+1 as our second, and also 2k+3 as our 3rd consecutive integer.
Next, translate “three consecutive odd integers whose amount is -321” right into an equation and solve because that k.
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Take the worth of k i beg your pardon is -54 and also use it to recognize the three consecutive strange integers.
Finally, verify that as soon as the three consecutive odd integers -109, -107 ,and -105 are added, the amount is -321.