LECTURE 15: INDUCTION STRENGTHENED (
§
6.4)
I think I should have no other mortal wants, if I could always have plenty of music.
It seems to
infuse strength into my limbs and ideas into my brain. Life seems to go on without effort, when I am
filled with music. George Eliot (1819  1880)
1.
Strong Induction
Recall our infinite staircase. To climb it, we needed to do two things. First, climb the bottom step.
Second, use the fact that we climbed the
k
th step to climb the (
k
+ 1)st step.
But we are not using everything available to us. We didn’t only climb the
k
th step, we also climbed
all the previous steps. So we can strengthen our inductive case. We do this by not only assuming
P
(
k
);
but rather we assume
P
(
i
) for all
i
in the range 1
≤
i
≤
k
. We still show that we can then get
P
(
k
+1).
Why don’t we always do this strong induction? Because sometimes it just complicates things.
Example 1.
Let
A
1
= 1 and
A
2
= 3. Then let
A
n
= 2
A
n

1

A
n

2
when
n
≥
3. What sequence do
we get? We have
A
3
= 2
A
2

A
1
= 2
·
3

1 = 5. Also
A
4
= 2
A
3

A
2
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 Spring '11
 Smith
 Logic, Mathematical Induction, Inductive Reasoning, Natural number, Fibonacci number, Strong Induction

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