1.Prove that the square of any odd integer has the form 8m + 1 for some integer m.

2.Prove the following statement by mathematical induction:

Thank you So much, again, for any help on these

(2n + 1)2	= 4n2 + 4n + 1
	4n(n + 1) + 1.

Note that in n(n + 1) the factors are two consecutive integers, of which one is necessarily even. Therefore

n(n + 1) = 2m, for some m.

Combining gives

(2n + 1)² = 8m + 1.

2. The induction in this example is so standard, it is probably contained in any book where math induction is mentioned.
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