**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.