The Art of Proof pp 33-45 | Cite as

# Recursion

Chapter

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## Abstract

**Before You Get Started**. You have most likely seen sums of the form \(\sum\nolimits_{j = 1}^k {j = 1 + 2 + 3 + \cdots + k,}\) or products like \(k! = 1 \cdot 2 \cdot 3 \cdots k.\) In this chapter we will use the idea behind induction to *define* expressions like these. For example, we can define the sum 1+2+3+ ∙ ∙ ∙ +(*k*+1) by saying, if you know what 1+2+3+ ∙ ∙ ∙ +*k* means, add *k*+1 and the result will be 1+2+3+ ∙ ∙ ∙ +(*k*+1). Think about how this could be done; for example, how should one define 973685! rigorously, i.e., without using ∙ ∙ ∙ ? Find a formula for 1+2+3+ ∙ ∙ ∙ +*k*.

## Keywords

Base Case Induction Step Fibonacci Number Fibonacci Sequence Binomial Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Matthias Beck and Ross Geoghegan 2010