Sums of Consecutive Counting Numbers

by Mr. John Bookston

Definition:  A sum of consecutive counting numbers can start with any counting number add at least the next bigger counting number and stop adding after any number of consecutive integers.

Examples:  3+4 = 7                   7 is a consecutive sum of counting numbers

   1+2+3+4+5 = 15    4+5+6 = 15 and 7+8 = 15     15 is a consecutive sum in 3 distinct ways.

Investigation: 

For numbers that are consecutive sums, investigate patterns to predict the number of distinct ways larger numbers can be written as consecutive sums.  

Starters:       1 and 2 are not consecutive sums

Every odd integer starting with 3 can be written as a consecutive sum of two counting numbers:

                                      (n-1)/2  +  (n+1)/2  =  n       for every odd number, n.

Make a conjecture about which numbers are expressible as “the sum of 3 consecutive counting numbers”. 

Extension: Do the same for numbers that can be expressed as sums of 4,5 and 6 consecutive counting numbers.

Extension:  Consider the prime factorization of the counting numbers that cannot be expressed as a consecutive sum.  Make a conjecture as to which counting numbers less than 1000 fall into that category.

Enjoy.             

Good luck. 

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