Mathematical%20Proof

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• Mathematical Proof

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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
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When showing that pn1 is true it is useful to
note that
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LHS RHS for n 1, hence p1 is true
This concludes step 1 of our proof
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This concludes step 2 of our proof
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note that we use n2 because we assumed that pn
is true
hence pn1 is also true
This concludes step 3 of our proof
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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
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True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n
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Exam type question
a)
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True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n
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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
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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
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(No Transcript)
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True for p1, assumed true for pn also true for
pn1 hence by proof of induction true for all n