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April 8th, 2014, 04:01 AM   #1
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induction

Use the method of induction to prove that the sum of first n odd numbers is equal to n^2

My attempt:

for n = 1
1 = 1^2 = 1
So it is true for n = 1

Assume true for 1 + 3 + 5 +...+ 2n - 1 = n^2

Now try for (n + 1)th term

1 + 3 + 5 +...+ 2n - 1 + 2(n + 1) -1 = n^2 + 2(n + 1) - 1
= n^2 + 2n + 2 - 1
= n^2 + 2n + 1
= (n + 1)^2

So it is true for 1, n and n + 1
Therefore the sum of first n odd numbers = n^2

Is this the correct method? Am I missing something? thanks
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April 8th, 2014, 04:26 AM   #2
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You have not missed anything, your proof is perfect.
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April 8th, 2014, 04:27 AM   #3
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The proof is correct, but there is no reason to represent 2n + 1 as 2(n + 1) - 1. Simply

1 + 3 + 5 +...+ (2n - 1) + (2n + 1) = n^2 + 2n + 1 (by IH)
= (n + 1)^2.

Also, it's a good idea to indicate where exactly the induction hypothesis is used.
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