Sum of Consecutive Integers (March 7-10)

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Which natural numbers can be written as the sum of two or more positive consecutive integers?  This exercise has challenged mathematicians for centuries and clear patterns have formed.  However, only recently, in 1997, did was a proof developed for why powers of two such as 1,2,4,8,16,32...... could never be a sum of consecutive integers.

·      Sum means add (the sum of 2 and 3 is 5).

·      Consecutive numbers are numbers like 2 and 3, 8 and 9 and 10, and so on.

·      Zero is not a positive whole number.

·      Find all combinations where consecutive positive whole numbers add up to the numbers from 1 through 55.  For extra credit, keep going.

·      Look for patterns and methods of finding the sums.

·      Put an “X” on the line if no sums can be found.

The students worked with the attached pdf worksheet to find these solutions.

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Below are a few observations and patterns that you may see arise:

Powers of 2 like 1,2,4,8,16… are not the sum of  consecutive positive numbers.

Prime Numbers like 2,3,5,7,11,13,17,19,23….. are the sums of only two consecutive positive numbers (one solution).

Odd Factors –the number of solutions will equal the number of odd factors of the number not including the number itself  (for example, the number 15 has three odd factors [1,3,5] not including itself; therefore, there are three solutions of consecutive positive numbers that have a sum of 15 like 7+8  =  4+5+6     =      1+2+3+4+5

Divisible by 3 – numbers divisible by three are fun because they will always have a solution with three consecutive positive integers with the number divided by 3 answer in the middle (for example, the number 18 divided by 3 is 6; therefore, 5+6+7 is a solution for 18). Look for patterns when numbers are divisible by other odd numbers like 5 and 7.

The solutions to the first 55 positive whole numbers are:

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1               =    X

2               =    X

3               =    1+2

4               =    X

5               =    2+3

6               =    1+2+3

7               =    3+4

8               =    X

9               =    4+5  =  2+3+4

10           =    1+2+3+4

11           =    5+6

12           =    3+4+5

13           =    6+7

14           =    2+3+4+5

15           =    7+8  =  4+5+6     =      1+2+3+4+5

16           =    X

17           =    8+9

18           =    5+6+7  =  3+4+5+6

19           =    9+10

20           =    2+3+4+5+6

21           =    10+11 =  6+7+8 =  1+2+3+4+5+6

22           =    4+5+6+7

23           =    11+12

24           =    7+8+9

25           =    12+13  =  3+4+5+6+7

26           =    5+6+7+8

27           =    13+14  =  8+9+10  =  2+3+4+5+6+7

28           =    1+2+3+4+5+6+7

29           =    14+15

30           =    9+10+11  = 6+7+8+9  =  4+5+6+7+8       

31           =    15+16                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              

32           =    X

33           =    16+17  = 10+11+12  =  3+4+5+6+7+8

34           =    7+8+9+10

35           =    17+18  =  5+6+7+8+9  =  2+3+4+5+6+7+8

36           =    1+2+3+4+5+6+7+8 = 11+12+13

37           =    18+19

38           =    8+9+10+11 

39           =    19+20  =  12+13+14  =  4+5+6+7+8+9

40           =    6+7+8+9+10

41           =    20+21

42           =    13+14+15  = 9+10+11+12  =  3+4+5+6+7+8+9

43           =    21+22

44           =    2+3+4+5+6+7+8+9

45           =    22+23  = 14+15+16  = 7+8+9+10+11  =  5+6+7+8+9+10  =  1+2+3+4+5+6+7+8+9

46           =    10+11+12+13

47           =    23+24

48           =    15+16+17

49           =    24+25  = 4+5+6+7+8+9+10

50           =    11+12+13+14 = 8+9+10+11+12

51           =    16+17+18  = 6+7+8+9+10+11 = 25 + 26

52           =    3+4+5+6+7+8+9+10

53           =    26+27

54           =    12+13+14+15  = 17+18+19

55           =    27+28  = 9+10+11+12+13  =  1+2+3+4+5+6+7+8+9+10

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