Alvin's Theorem

     While Ms. Powers was leading a class discussion about square numbers, Absent-minded Alvin was in another "world", looking for interesting patterns in the topic. Shortly, he raised his hand and said, "Ms. Powers, I've found something rather nice. Look. If I take 2 consecutive squares and subtract them, the difference is always the sum of 2 consecutive integers."

     "Show the class what you mean by that, Alvin," said the teacher.

     Alvin wrote the following on the board:

49 - 36 = 13 and 13 = 6 + 7

64 - 49 = 15 and 15 = 7 + 8

     Turning to the class, he shyly said, "I call this 'Alvin's Theorem'."

     Ms. Powers smiled and said, "Very good, but if you want to call it a theorem, you must be able to prove it is always true for all numbers, using algebra."

     Alvin replied, "Oh yes, I can do that too. Here's how."

     What did Alvin write on the board now?

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     Extra: Later, Alvin investigated the matter of the difference of consecutive "even" squares. What do you imagine he discovered this time?