M1410904 - (Chapter 9: Discrete Math) 9.26 SECTION 9.4:...

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(Chapter 9: Discrete Math) 9.26 SECTION 9.4: MATHEMATICAL INDUCTION What is the sum of the first n positive integers, where n Z + ? In other words, what is 1 + 2 + 3 + ... + n ? According to our formula for the n th partial sum of an arithmetic sequence (see Section 9.2 ), the answer is: S n = n 1 + n 2 = n n + 1 ( ) 2 Handshake Problem (Cool, but Optional) There’s another way of seeing why this formula works out. Imagine n + 1 people walking into a room one-by-one. Whenever a person walks into a room, he/she must shake hands exactly once with every other person who is currently in the room, and those are the only handshakes they make. The first person who walks into the room shakes nobody’s hand, the second person shakes the first person’s hand, the third person shakes the first two people’s hands, and so on, until the last person shakes the other n people’s hands. This means that every distinct pair of people eventually shake hands exactly once. The total number of handshakes then equals the number of distinct pairs of people that can be formed from a group of n + 1 people.
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M1410904 - (Chapter 9: Discrete Math) 9.26 SECTION 9.4:...

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