The sum of two or more consecutive integers

HOME

Math Central

Quandaries & Queries

Question from nana, a student:

some numbers can be written as the sum of two or more consecutive integers (we consider positive integers only)
other number (eg. 4) cannot be express in this way.
let us call a number which can be expressed in this way a SOALTCI( sum of at least two consecutive integers)
a). list the first few SOALTCI and conjecture the general formula.
b). Prove that any number of the given form(in your answer (a)) is a SOALTCI

Nana,

I'll give you a start - I'm guessing you've tried the first few numbers already and noticed that 2,4 & 8 don't work? What's special about these numbers?

Consider 100 = (2x12+1)(4) = (12-3)+(12-2)+...+12+13+...+(12+4), a sum of 8 numbers;

also consider 320 = (2x2+1)(64) = 62+63+...+66 a sum of 5 numbers.

What's the pattern?

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.