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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 3-4, Math Kangaroo 5-6.
Consecutive numbers are integer numbers that form a sequence increasing or decreasing by 1.
Several easy but useful facts about the consecutive integers:
Fact #1: A product of two consecutive integers is always a multiple of 2 (is even, as one of them is bound to be a multiple of 2.)
Fact #2: A product of three consecutive integers is always a multiple of 3 (one of them is bound to be a multiple of 3).
Fact #2: N consecutive numbers starting at 1 add up to:
An easy proof of this fact is a variation of the one attributed to Karl Friedrich Gauss:
By adding the terms in each vertical column we notice that each column sums up to the same number: (N+1). There are N columns in total, therefore the grand total of the sum is N(N+1). However, this represents twice the sum S which we are asked to compute. Therefore,
Fact #3: It is often useful to symmetrize odd numbers of consecutive integers with respect to the middle number M, like this:
Fact #4: The average of an odd number of consecutive numbers is the middle number.
Prove this using a symmetrized sequence like above. There are 2k+1 numbers in this sequence:
and their average is:
We notice that all the numbers added to M cancel out with numbers subtracted from M leaving only 2k+1 Ms: