Lesson 8

# Lesson 8 - Induction Imagine you have a staircase We need...

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Induction Imagine you have a staircase. We need two things in order to be able to walk up it, no matter how long it is. The ﬁrst step must be safe. Also every step after a safe step must be safe. Mathematically we need two things for a proof. We need to establish that the ﬁrst (step 1) step is true. Also, assuming a step is true (step k), the next step must be true(step k+1). We call these the base case and the inductive step . Let us do this example: 1 + 2 + ... + n = n ( n + 1) 2 For the base case, we need to verify that the statement is true for n = 1. LHS = 1 RHS = (1 * 2) / 2 = 1 For the inductive step, we assume that what we want is true for n = k . This means: 1 + 2 + ... + k = k ( k + 1) 2 Now we need to show the statement for n = k + 1: 1 + 2 + ... + k + ( k + 1) = ( k + 1)(( k + 1) + 1) 2 = ( k + 2)( k + 1) 2 Basically, we start with the left side and use our assumption to get the right side. LHS

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## This note was uploaded on 03/22/2010 for the course MATH 1104 taught by Professor Unknown during the Fall '10 term at Carleton.

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Lesson 8 - Induction Imagine you have a staircase We need...

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