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Consider what a consecutive number means here. If you start with a number 2, it is obvious that the consecutive numbers to this are 3, 4, 5, 6, etc. But how do you get to these numbers based on the value of 2?
3 = 2 + 1
4 = 2 + 2
5 = 2 + 3
etc.
Now if you replace this number with an unknown variable 'n', you can figure out this question. If you have some number 'n', a consecutive number to this will be n+1, n+2, etc. just as it was with my example with 2.
This problem is a little trickier because you do not know how many consecutive numbers there are, and for this reason there are multiple answers, but I will go through all of them.
In order to think about this, let's explore another example.
Say you have the problem: which four consecutive numbers add up to 18?
You would go about this with n, n+1, n+2, and n+3. The reason for the parenthesis is to help with organization; each term is a different consecutive number. Then, we simplify.
n + (n+1) + (n+2) + (n+3) = 18
n + n + 1 + n + 2 + n +3 = 18 Simplification
n + n + n + n + 1 + 2 + 3 = 18 Commutative Property of Addition
4n + 6 = 18 Combining like terms
4n + 6 - 6 = 18 - 6 Subtraction Property of Equality
4n + 6 - 6 = 18 - 6 Simplification
4n = 12 Subtracting like terms
4n/4 = 12/4 Division Property of Equality
n = 3 Simplification
So, now we know the first number, so we can find the other numbers from this.
n, n+1, n+2, n+3
3, (3+1), (3+2), (3+3)
3, 4, 5, 6
You can check your answer if you would like.
3 + 4 + 5 + 6 = 18
There is one interesting thing to note about this problem - what is the average?
The average is 18/4, or 4.5
The average of an even number of consecutive numbers will always be halfway between the middle two numbers. In other words, the average will never be an integer. This is always true (you can test it out if you like).
If we added 7 to this, let's reevaluate the average to see what happens for sums of odd numbers of consecutive numbers.
3 + 4 + 5 + 6 + 7 = 25
25/5 numbers = 5
Here, the average is an integer, and this will always be the case (feel free to test this too).
These two pieces of information, that the sum of an even number of consecutive numbers will not be an integer (will end in 0.5, actually) , and that a sum of an odd number of consecutive numbers will be an integer, we can now solve your problem, along with some investigation.
Now, because the number of consecutive numbers must be an integer, we'll divide 50 by integers until we get an average that fits with our previous discoveries.
We will look for even divisors whose quotients end in 0.5, and for odd divisors whose quotients are integers.
50/1 = 50
50/2 = 25
50/3 = 16.66666...
50/4 = 12.5
50/5 = 10
50/6 = 8.333...
50/7 = 7.142857142857...
50/8 = 6.25
50/9 = 5.55555...
50/10 = 5
You might wonder why I knew to stop at ten. Because 1+2+3+4+5+6+7+8+9+10 =55, which is greater than our sum, and these are the lowest positive consecutive integers, more than ten positive consecutive integers is not possible.
So, we have two answers, assuming that all consecutive numbers are positive.
50/4 = 12.5
(Same steps as the first example)
n+n+1+n+2+n+3=50
4n+6=50
4n=44
n=11
11, 12, 13, 14
50/5 = 10
(almost same steps)
n+n+1+n+2+n+3+n+4=50
5n+10=50
5n=40
n=8
8, 9, 10, 11, 12
Now, if you are allowed to also use negative numbers, there are two more answers. Since 1+2+3+4+5+6+7 plus each number's opposite is zero, we can add all of these numbers, their opposites, and zero, and this will not affect the sum (they are still consecutive numbers)
-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
If assuming only positive integers:
11, 12, 13, 14
8, 9, 10, 11, 12
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What are four consecutive numbers that add up to 268?
There are no such numbers. If S is the sum of any 4 consecutive integers then S = 2 (mod 4) In other words, any four consecutive integers add up to an even number that is NOT dvisible by 4.
The sum of 2 consecutive integers is less than 55 find the pair of integers with the greatest sum?
The pair of consecutive integers which add up to 55 are 27 and 28. Therefore, any consecutive pair of numbers below 27 and 28 add up to a total less than 55.
What three consecutive numbers add up to 25?
That isn't possible. The three consecutive number are assumed to be integers; the sum of three consecutive integers is always a multiple of 3 (try it out).
What are two consecutive numbers that add up to 600?
The sum of consecutive integers will always be odd. Consecutive odd numbers will be even. 299 + 301 = 600
What are three consecutive numbers that add up to 66?
The average of three numbers which sum to 66 is 66/3 = 22. If 22 is the sum of three consecutive integers, then those three integers are 21,22,23. Kermit Rose
What is the sum 98 find the smaller number The consecutive integers?
No two consecutive integers can add up to 98.No three consecutive integers can add up to 98.But 23, 24, 25, and 26 can.
What are four consecutive numbers that add up to 36?
This is impossible - no four consecutive integers add to 36.
What are three consecutive odd integers that add up to -27?
The integers are -11, -9 and -7.
What is three consecutive even integers whose sum is 36?
10 12 14 are the consecutive even integers which add up to 36
The sum of four consecutive even integers is the same as the least of the integers Find the integers?
-4, -2, 0, and 2 are the four consecutive even integers. When you add them up they equal -4.
What are two consecutive integers add up to 26?
This is impossible, in mathematical terms. If you take two consecutive integers, then one of the integers must be odd and the other must be even. When you add an odd number to an even number, the result is always an odd number. Here, you said two consecutive integers add up to 26, which is an even number. Therefore, the answer is "No real solutions."
The sum of 2 consecutive integers is less than 55 find the pair of integers with the greatest sum?
The pair of consecutive integers which add up to 55 are 27 and 28. Therefore, any consecutive pair of numbers below 27 and 28 add up to a total less than 55.
What are four consecutive numbers that add up to 268?
There are no such numbers. If S is the sum of any 4 consecutive integers then S = 2 (mod 4) In other words, any four consecutive integers add up to an even number that is NOT dvisible by 4.
What three consecutive numbers add up to 25?
That isn't possible. The three consecutive number are assumed to be integers; the sum of three consecutive integers is always a multiple of 3 (try it out).
What are three consecutive integers that add up to ninety?
29, 30, and 31
What is the least number of consecutive integers that add up to 1000?
You need 5.
What two consecutive integers add up to -41?
The numbers are -21 and -20.
