General Methods for Finding a Closed Form (Methods 0 - Look it Up and 1 - Guess & Prove)

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Based on the book, Concrete Mathematics, by Graham, Knuth, and Patashnik. (Chapter 2.5) Given a summation - how do you find a closed form? Method 0 - Look it up. Method 1 - Guess and Prove by Induction.
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For method 0, how did you find that answer out? I understand the formula is right, but mathematically, how did you get to that conclusion? Thank you.

catalindeluxus
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Wow, I forgot cursive writing existed lol.

kurchak