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