# Math Research Project — Sums of Integers

Each Math Research Project outline consists of a *Seed Question*, *Questions to Build On*, *Extensions*, and *Basic Background*. Click here to see a list of available Math Research Project outlines.

The *Seed Question *is a question that appears in a typical math curriculum. *Questions to Build On* are simple extensions, re-interpretations, and generalizations of the* Seed Question *that a student can build a simple math research paper around, with help from a mentor (be it student or teacher).

The *Extension Questions *are bigger, more challenging extensions of the *Seed Question*. If a student can make some headway into understanding these, great! If not, they are good questions to include as part of the “Where do we go from here?” section of the paper.

The *Basic Background *consists of some of the mathematical leg-work that underlies the investigation. It can be worked on simultaneously with the investigation, which gives it context

### Sums of Integers

Find the sum of the first 100 consecutive positive integers: 1 + 2 + 3 + … + 100.

**Questions to Build On**

Find the sum of the first 1,000 consecutive positive integers: 1 + 2 + 3 + … + 1000.

Find the sum of the first *n* consecutive positive integers: 1 + 2 + 3 + … + *n*.

Find the sum of *n* consecutive positive integers, starting from *k.*

**Extension Questions**

Find a series of consecutive integers that add up to 5015.

Find all series of consecutive integers that add up to 5015.

What integers could be the sum of consecutive integers?

Find the sum of the first 100 perfect squares: .

Find the sum of the first *n triangular numbers*: 1 + 3 + 6 + 10 + 15 + …. + *n(n-1)*/2.

**Basic Background**

Prove that *Gauss’s Trick* works.

Derive the formula for the sum of an arithmetic series in two different way.

*Click here to see a list of available Math Research Project outlines.*