Mathematical Induction

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Mathematical Induction
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Consider the following statement

• Any n squares can be cut into pieces and then
  pasted into one single square.
• Let us denote this statement by S(n).
• Of course n ? 2.
• First of all, let us consider S(2)
• (please refer to 2squares.gsp)

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Conclusion

• Hence S(2) is true, that is, any 2 squares can be
  cut into pieces and then pasted into one single
  square.
• But how about other statements?
• So, lets go on to S(3).

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? S(3) is true.
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? S(4) is true.
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? S(5) is true.
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Conclusion

• Hence, we can see that S(n) is true for all
  integers n ? 2 because
• 1. S(2) is true,
• 2. If S(k) is assumed to be true,
• then S(k1) is true.

S(2)
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Principle of Mathematical Induction

• Let S(n) be a statement about a positive integer
  n ? 1. Then S(n) is always true if
• 1. S(1) is true.
• 2. If S(k) is assumed to be true,
• then S(k 1) is true.

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Example

• To prove For all n ? 1,
• 1 2 3 n

Proof When n 1, L.H.S. 1 R.H.S.
?L.H.S. R.H.S. when n 1.
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• Next, assume that LHS RHS when n k,
• that is,
• 1 2 3 k

Then, when n k 1, LHS 1 2 3 k
(k 1)
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Conclusion

• Hence, by principle of M.I., LHS RHS for all n
  ? 1.

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Worksheet

• Show that for any natural number n,
• 12 22 32 n2