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Published in:
2010 | OriginalPaper | Chapter
Recursion
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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
.