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induction

# induction - MATH 204 PROOF BY INDUCTION MRINAL RAGHUPATHI...

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MATH 204. PROOF BY INDUCTION. MRINAL RAGHUPATHI These notes are an informal introduction to proof by induction. A more careful presentation of these ideas may be given during the course. To motivate this technique we begin with a simple problem. What is the sum of the first n natural numbers? What we are looking for is a formula for the sum 1 + 2 + · · · + n . This number arises for instance when you want to count the number of handshakes that are required at a business meeting where there are n people, or the number of “clinks” at a toast. Here is an elegant solution due to Gauss. Write the numbers from 1 to n in reverse and add: 1 2 · · · n + n ( n - 1) · · · 1 = n + 1 n + 1 · · · n + 1 This calculation shows that 2(1 + 2 + · · · + n ) = n ( n + 1) from which we get 1 + 2 + · · · + n = n ( n + 1) 2 . We may not be as clever as Gauss. A good starting point is to compute the sum for some small values of n and look for a pattern. Here are the first ten values: (1) 1 , 3 , 6 , 10 , 15 , 121 , 28 , 36 , 45 , 55 .

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