Recursion

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.