# If you add all the consecutive numbers is one of the totals 1000?

No. There is no number n, such that the sum of the first n consecutive integers is 1,000. Here's why: Let S = S(n) = 1 + 2 + 3 + . . . + n. Then, S = n(n + 1) / 2. There are several ways you can derive this formula; but you might find it helpful to refer to this: http://mathworld.wolfram.com/ArithmeticSeries.html or any of a large number of similar sources. (I tried Wiki, but down-loading took forever!) If you know S(n) (in this case, 1,000) and wish to find n, you have only to solve: n² + n = 2S = 2000. This is not readily factorised; but we can try completing the square: 2S = n² + n = 2,000; whence, n² + n + ¼ = (n + ½)² = 2,000¼, and n = √2,000.25 - 0.5. To the nearest integer, n = 44, approximately; but this is inexact, of course. Let's see what happens when we try this result. We get 2S(44) = n² + n = 44 * 45 = 1,980; whence, S(45) = 990, which is just a little too small; and any n smaller than 44 would give an S(n) that is even smaller. You can easily check this result, if you like: 1 + 2 + 3 + . . . + 22 + 23 + . . . + 42 + 43 + 44 = (1 + 44) + (2 + 43) + (3 + 42) + . . . + (22 + 23) {this makes 22 pairs, each adding up to 45.} = 45 * 22 = 990. On the other hand, S(45) = S(44) + 45 = 990 + 45 = 1,035, which is just a little too large; and any n larger than 45 would give an S(n) that is even larger. We may conclude, then, that no integer n satisfies the requirement that we have placed on it.

### What are four consecutive numbers that add up to 35?

There are no four consecutive whole numbers that add up to 35. The sum of two consecutive [whole] numbers is an even number plus an odd number which is an odd number. The sum of two consecutive numbers and the two next consecutive numbers is the sum of two odd numbers which is even, but 35 is odd, so no four consecutive whole numbers cannot add up to 35.

### What are two pairs of consecutive numbers that add up to 5280?

Although not specified as such, "consecutive" requires the numbers to be integers. Two pairs literally means four numbers but there are not four consecutive integers that add up to to 5280. 1318 + 1319 + 1320 + 1321 = 5278 and 1319 + 1320 + 1321 + 1322 = 5282. Four consecutive numbers must add up to an even number that is not a multiple of 4. If by two pairs, the question meant ONE…