Question 1039711: show how 9 can be expressed as the sum of two consecutive numbers. then decide whether every odd number can be expressed as the sum of two consecutive counting numbers. explain your reasoning.
Answer by Theo(10140) (Show Source):
You can put this solution on YOUR website!
1 + 2 = 3
2 + 3 = 5
3 + 4 = 7
4 + 5 = 9
5 + 6 = 11
it looks like any consecutive numbers will be composed of one even number and one odd number, therefore their sum will have to be odd.
conversely, if the number is odd, then it has to be composed of 2 consecutive counting numbers.
all except 1, which is not composed of anything except 1 and 0, but 0 is not a counting number.
therefore, the smallest counting number is 1 which stands by itself.
any other counting number that is odd, can be composed of two consecutive counting numbers.
divide it by 2 and you get 5 + a remainder of 1.
so 11 is equal to 2 * 5 + 1 which can also be shown as 5 + (5 + 1) which can be shown as 5 + 6 which are consecutive numbers.
divide it by 2 and you get 6 + 1.
so 13 is equl to 2 * 6 + 1 which can also be shown as 6 + (6 + 1) which can be shown as 6 + 7 which are consecutive numbers.
the first number in the sequence can be odd or even.
the second number in the sequence will be odd if the first number is even, and will be even, if the first number is odd.
add any odd number and any even number together and you'll get an odd number.
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