1 / 6

The Most Important Sum

The Most Important Sum. 1+2+3+…+(n-1)+n = . n(n+1) 2. Applications:. n people; how many handshakes ? (equivalent: n cities to directly connect with an internet cable; how many cables needed? ) Arranging n cards in order: how many compares ?

talor
Download Presentation

The Most Important Sum

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. The Most Important Sum 1+2+3+…+(n-1)+n = n(n+1) 2

  2. Applications: • n people; how many handshakes? • (equivalent:ncities to directly connect with an internet cable; how many cables needed?) • Arranging n cards in order: how many compares? • Appending to a list of names: how many steps? • Keeping students busy (anecdote: Gauss)

  3. Three derivations • Gauss' approach: 1 + 2 + 3 + … + (n-2) + (n-1) + n = S n +(n-1)+(n-2)+ … + 3 + 2 + 1 = S (n+1)+(n+1)+(n+1)+…+(n+1) + (n+1) + (n+1) = 2S n(n+1) = S 2

  4. Three Derivations (part 2) 1 2 3 . . . n-1 n . . . . . . . . . 1 2 3 n-1 n

  5. Three Derivations (part 2) . . . 1 2 3 . . . n-1 n . . . . . . . . . . . . n(n+1) = S 2 . . . 1 2 3 n-1 n

  6. Three Derivations (part 3) Count the ways to create a handshake: • Choose the 1st person (n choices) • Choose the 2nd person (n-1 remaining choices) • That's n(n-1) ways to choose one then the other. • But this overcounts: "Choose Alice then Bob" and "Choose Bob then Alice" are the same handshake; divide by 2 to correct. S = n(n+1) 2

More Related