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## Etude for the Number N

Consider the number 3 and write it as a sum:

1 + 1 + 1 = 3
1 + 2 = 3
2 + 1 = 3
3
Conclusion: The number 3 can be written as the sum of one or more natural numbers in four ways. Note that order matters, and the commutivity aspects of addition are ignored.

Your Task: Determine how many ways the natural number N can be expressed as a sum of one or more natural numbers? Provide a justification as well.

Note: Why does it not make any sense to ask how many ways the natural number N can be expressed as a sum of one or more integers?

Source: W. Fleming's College Algebra, 1988, p. 416

Hint: Start small...work out the cases for 1, 2, 4, etc. Do you see patterns....in both the result and how that result is obtained or justified.

Solution Commentary: It is fairly straight forward (exhaustion by effort only) to build a conjecture about the result...but building an argiment as to why the pattern holds--that is another story.

This specific area of mathematics is called "compositions." To check your answer and argument, you might consider this Wikipedia site. The proof provided is very creative...you can learn alot by studying it....