## INVESTIGATION EXAMPLE

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### CONSECUTIVE SUMS

Note that the natural number 9 , for example, can be written as a sum of consecutive natural numbers (positive integers) in two different ways:

9 = 4 + 5
and
9 = 2 + 3 + 4

Your task is to explore such "consecutive sums" for all natural numbers (not just 9).

#### Patterns

Take each natural number from 1 to 30 and:

* Find all the ways each number can be written as a sum of consecutive natural numbers.

* Record these ways in some meaningful form in a chart.

* Search for as many patterns as you can find in your results.

* Write summary statements describing each pattern.

* For each pattern you found, try to explain why it holds.

#### Generalizations

Address as many of the following generalizations as you can. Partial explanations are better than none, but make your answers as complete, clear, and direct as you can.

* Describe the impossible numbers, that is, natural numbers which cannot be written as a consecutive sum.

* Find the "triangular numbers" somewhere in your chart and explain how they fit into the scheme of consecutive sums.

* Find all possible consecutive sums for 198 .

* Given ANY natural number, describe a procedure for finding all possible runs summing to it. How do you know when you've found them all?

* How does the average (mean) of all the numbers in a given run relate to your findings?

* For a given natural number, look at its factors and try to relate them to its associated runs. Try to use this to predict how many different runs sum to any given number and to describe a method for finding them.

#### PRESENTATION

Write a single group report showing your chart, describing and explaining the patterns you found, and addressing the generalizations above (as far as you can). The report can take any form or length you see fit, but it must address the problem adequately and be well-written and neat.

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