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).
PROBLEM
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 wellwritten and
neat.
