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Mathematical Proof

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Mathematical Proof A mathematical statement is either true or false Proof by induction can be used to show that a given statement is true for all n Step1 Step2 Step3 ... – PowerPoint PPT presentation

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Title: Mathematical Proof


1
  • Mathematical Proof

2
A mathematical statement is either true or false
Proof by induction can be used to show that a
given statement is true for all n
3
When showing that pn1 is true it is useful to
note that
4
LHS RHS for n 1, hence p1 is true
This concludes step 1 of our proof
5
This concludes step 2 of our proof
6
note that we use n2 because we assumed that pn
is true
hence pn1 is also true
This concludes step 3 of our proof
7
True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n
This concludes step 4 of our proof
8
True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n
9
Exam type question
a)
10
True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n
11
We can therefore use the summation notation
and the statement pn can be rewritten
This can be shown to be true for all n by
using the four steps of proof by induction
12
Proof by induction that 4n -1 is divisible by 3
for all positive integers n
True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n
Alternatively
13
(No Transcript)
14
True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n
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