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Trang 1Induction
Trang 2The Idea of Induction
0, 1, 2, 3, 4, 5, …
then you know that
all the ints are red !
Trang 3Induction Rule
R ( 0 ), R ( 1 ), R ( 2 ),…, R ( n ),…
R ( 0 )
and (" n) ( R ( n )fi R ( n+1 ))
(" m) R ( m )
Trang 4Like Dominos…
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Trang 5Example Induction Proof
Let’s prove:
r-1
(for r ≠ 1)
Trang 6Statements in magenta form a
template for inductive proofs:
• Proof: (by induction on n )
• The induction hypothesis, P( n ) , is:
Example Induction Proof
1+r +r + +r =
r-1 L
(for r ≠ 1)
Trang 7Base Case (n = 0):
r - 1
= =
r - 1 1
L
0+1
1+r +r + +r =
r - 1
Example Induction Proof
1
OK!
Trang 8• Inductive Step: Assume P ( n ) for some n ≥ 0
and prove P ( n+1 ) :
Example Induction Proof
+1
n+ 1 r ( ) - 1
r - 1 L
Trang 9Now from induction
hypothesis P(n) we have
1+r +r + +r =
r - 1
Example Induction Proof
so add rn+1 to both sides
Trang 10adding rn+1 to both sides,
n+1 n+1
+1
(n+1)
r - 1+r (r - 1)
=
r - 1
r - 1
=
r - 1
r - 1
L
Example Induction Proof
This proves
P(n+1)
completing the
proof by induction.
Trang 11“” is an ellipsis.
• Can lead to confusion (n = 0 ?)
• Sum notation more precise
Means you
an aside: ellipsis
∑n i i=0
L
1+r +r + +r