How many positive integers from to inclusive can be written as the sum of two or more consecutive positive integers? (Different examples and : and )

Source: mathcontest.olemiss.edu 3/7/2011

**SOLUTION**

**Sum of two consecutive numbers**

**Sum of three consecutive integers**

duplicates are indicated by underlines.

**Sum of four consecutive integers**

**Sum of five consecutive integers**

**Sum of six consecutive integers**

**Sum of seven consecutive numbers**

**Sum of eight consecutive integers**

**Sum of nine consecutive integers**

There are seven numbers in this category but they are duplicates.

**Sum of ten consecutive integers**

There are five numbers in this category but they are duplicates.

**Sum of eleven consecutive integers**

There are four numbers in this category; all are duplicates except . Thus, there is one number that can be written as the sum of eleven consecutive positive integers.

**Sum of twelve consecutive integers
**There are two numbers in this category; both are duplicates.

**Sum of thirteen consecutive integers**

There is one number in this category; it is a duplicate.

**Sum of fourteen consecutive integers
**.

The sum is greater than . So, we stop here.

Total =

**Answer**:

**Alternative solution**

Let be the sum of consecutive integers with the first term equal . We apply the sum formula for various values of .

is an odd integer. The integers are .

The odd integers are duplicates of case . The remaining even integers are .

The multiples of are duplicates of case . The remaining integers are .

Fourteen of the integers are duplicates of case and . The remaining integers are .

All the integers are odd and duplicates of case .

Nine of the integers are duplicates. The remaining integers are .

Four of the integers are duplicates. The integers are .

All the integers are duplicates.

All the integers are duplicates.

All the integers are duplicates except .

Both integers are duplicates.

is a duplicate.

so we stop.