Proof by Induction
1.
2.
3.
4.
5.
Explanation 1
Explanation 2
Example Division
Example Sequences
Example Factorials
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Is every car in the line red?
This would constitute a proof or would it?
1.Find one car that is red
2.Show that every car is the same as the one beside it
These are the basic steps in the Mathematical
proof by induction
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Mathematical steps
It May be necessary to
use a number greater
1 on some
Proof by induction we will apply tothan
statements
that have a sequential element
occasions
1. Prove statement is true for 𝑛 = 1 where n is a Natural number.
2. Assume that the statement is true for some Natural number K
3. Using this assumption prove the statement true for k + 1
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For example Prove that numbers of the form 6𝑛
− 1 are always divisible by 5
∀𝑛 ∈ 𝑁, 𝑛 ≠ 0
Proof by Induction
Let 𝑛 = 1
61 −1
Therefore
5
5
5
= =1
Correct
6𝑘 −1
Assume true for k , this means
∈𝑁
5
Consider whether 6𝑘+1 − 1 is divisible by 5
𝑘+1
−
6
Take a bit of
each!
1
6. 6𝑘 −6𝑘 . 1 + 6𝑘 . 1
6𝑘
6−1 +
1(6𝑘
−
1
Factorise
− 1)
ThereforeTrue by
∴
Obviously
6𝑛 −true
1 are
assumption
always
divisible by 5
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Prove by Induction that 1 + 3 + 5 + 7 + ⋯ + 2𝑛 − 1 = 𝑛2
Let 𝑛 = 1, that gives 1 = 1
This is
Correct
the
nth
odd
number
2
Assume true for k that is
1 + 3 + 5 + 7 + ⋯ + 2𝑘 − 1 = 𝑘 2
If you add n odd
numbers the answer
is n2
Consider the k + 1 case
+
2 𝑘 + 1 − 1)
1 + 3 + 5 + 7 + ⋯ + 2𝑘 − 1 +
= 𝑘(2
(2 𝑘 + 1 − 1)
the1 same
1 + 3 + 5 + 7 + ⋯ + 2𝑘 − 1 + 2 𝑘 +Add
1 −
= 𝑘 2thing
+ 2𝑘to+both
2−1
1 + 3 + 5 + 7 + ⋯ + 2𝑘 − 1 + 2 𝑘 + 1 − 1 =sides
𝑘 2 + 2𝑘 + 1
1 + 3 + 5 + 7 + ⋯ + 2𝑘 − 1 + 2 𝑘 + 1 − 1 = (𝑘 + 1)2
So now if you add K + 1 odd number the answer is (𝑘 + 1)2
The proposition is proved
Quid erat demonstrandum
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Prove by Induction that 𝑛!
> 3𝑛−1
∀𝑛 ≥ 5, n ∈ 𝑁
Note here, the statement is only true when n is at least 5
Let 𝑛 = 5
5! > 35−1
120 > 81
Correct
Assume true for k and consider for k + 1
Is true
𝑘! > 3𝑘−1
𝑘 + 1 ! > 3𝑘+1−1
𝑘 + 1 . 𝑘! > 3. 3𝑘−1
We know, by assumption that 𝑘! > 3𝑘−1 ,
so this statement must true if 𝑘 + 1 ≥ 3
Which means
𝑘≥2
But 𝑘 ≥ 5
So it must be true for k + 1
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