Math, asked by jocelynd1102, 6 months ago

The sum of six consecutive integers is −9. What are the integers?

Answers

Answered by Anonymous

Since the six numbers total 183, it means that they average 30.5 (183/6). Since the numbers are sequential — hence incrementally spaced by a difference of one — and 30.5 represents the “middle” of this six digit sequence, this means that the three whole numbers immediately before and the three whole numbers immediately after the average value of 30.5 constitute the sequence. So the six consecutive numbers are: 28, 29, 30, and 31, 32, 33, which, indeed total to 183, and the largest, obviously, is 33.

Another way of looking at it is that, for a consecutive sequence of any even number of digits (say, X), there are X/2 pairs which total to the same number, and that number will be equal to the sum of all the numbers divided by the number of pairs (X/2). For the example at hand, that means there are 3 pairs of numbers that each total 61 (from 183/3), and which are centered around the average of 30.5. In this case, that means the pairs are 30 and 31; 29 and 32, and 28 and 33, which again yields the sequence 28–33.

Both of these methods derive from a simple but powerful concept deduced by the famous mathematician Gauss when he was only an elementary school student. The gist of the story is that the teacher assigned the class the task of coming up with the sum of the numbers 1 through 100. Rather than laboriously adding the numbers together, Gauss quickly realized that this large sequence consisted of 50 pairs (1+100, 2+99, 3+98, etc.), all totaling 101. In this way he was able to come up with the sum of 5050 (50 X 101) within a matter of seconds, astonishing his teacher. [With a slight variation, this method also works if there isn’t an even number of consecutive numbers…you just have to determine the middle number in the sequence and temporarily set it aside. For example, to sum up the sequence of 1–101, the middle number is 51 (1 +101, divided by two). The remaining 100 numbers are the 50 pairs: 1+101, 2+100, 3+99, etc., each equal to 102. So the sum of the sequence 1-101 is: 5100 (from 50 X 102), plus the lone “middle” number, 51, for a grand total of 5151.] And, hopefully, it’s obvious this method of summing works with any consecutive sequence of integers. The sequence does not have to begin at 1: it could be the sequence of, say, 317 through 5264; or -49 through -23; or -14 through +71 ; etc. You just have to determine the number of pairs, and whether the total sequence has an odd or even number of integers, and then apply one of the two methods above.