Sums of Consecutive Natural Numbers

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Presentation transcript:

Sums of Consecutive Natural Numbers Which natural numbers can be written as the sum of two or more consecutive natural numbers? Analyze by listing some examples.

1 = ? 17 = 8+9 2 = ? 18 = 3+4+5+6 or 5+6+7 3 = 1+2 19 = 9+10 4 = ? 20 = 2+3+4+5+6 5 = 2+3 21 = 10+11 or 6+7+8 or 1+2+3+4+5+6 6 = 1+2+3 22 = 4+5+6+7 7 = 3+4 23 = 11+12 8 = ? 24 = 7+8+9 9 = 4+5 or 2+3+4 25 = 12+13 or 3+4+5+6+7 10= 1+2+3+4 26 = 5+6+7+8 11= 5+6 27 = 13+14 or 8+9+10 or 2+3+4+5+6+7 12= 3+4+5 28 = 1+2+3+4+5+6+7 13= 6+7 29 = 14+15 14= 2+3+4+5 30 = 9+10+11 or 6+7+8+9 or 4+5+6+7+8 15= 7+8 or 4+5+6 31 = 15+16 or 1+2+3+4+5 32 = ? 16= ?

Maybe Gauss used algebra and 1 + 2 + 3 +. . .+100 = S the commutative property of addition 100 + 99 + 98 + . . .+ 1 = S 101+101+101+. . .+ 101 = 2S 100 (101) = 2S S = 100 (101) = 5,050 2 1. Can you find a formula for the sum of 1 + 2 + 3 + . . . + n using the commutative property

Did you come up with + 2 + 3 + . . . + n = S n + (n-1) + (n-2) . . . + 1 = S (n+1) + (n+1) + (n+1) + . . + (n+1) = 2S n (n+1) 2 or n is the # of terms and n+1 is the sum of the 1st & last terms Since the last term in a series will not always be equal to n, the # of terms, and the first term in a series is not always 1, let’s rewrite the formula n (first # + last #) 2 Can you find the sum of 72 + 73 + 74 + . . . + 99 using this formula?

Did you come up with 28 (171) = 4788 = 2,394 2 2 Was there a tricky part to this problem? How did you solve it? What if you have to find the sum of consecutive odd numbers from 1 to 101?

How many numbers were we adding together? How many terms? 1 + 3 + 5 + 7 +. . . + 101 Did you add 1 and divide by 2 to get 51 terms? Did you know that 51 is 2,601, the answer that you should have come up with?

Take a look at these sums to see why = 1 = 1 1 + 3 = 4 = 2 1 + 3 + 5 = 9 = 3 1 + 3 + 5 + 7 = 16 = 4 1 + 3 + 5 + 7 + 9 = 25 = 5 The sum of odd numbers are perfect squares.