The sum of consecutive positive integers is 1998. If is odd, what is the greatest possible value for ?

Source: mathcontest.olemiss.edu 9/26/2011

**SOLUTION**

Before tackling this problem we should revisit two formulas related to the sum of consecutive positive integers. Let where is odd be a collection of consecutive positive integers.

For example, is such a collection where

**Sum formula**

The formula that calculates the sum is given by

The second factor in the above formula really is the number of terms in the collection, so in this problem we have a simpler formula

The sum in our example

**Sum of first and last term
**

Thus,

(1)

In our example,

**Guess and check method**

Applying the sum formula yields

Decomposing 1998 into prime factors gives

**Case 1**:

Applying Eq. (1)

not a solution because negative integers are not allowed.

**Case 2**:

Applying Eq. (1)

not a solution

**Case 3**:

Applying Eq. (1)

not a solution

**Case 4**:

Applying Eq. (1)

a solution!

**Check
**

**Answer**: .