Comments on: Mathematical structures, 5 http://cameroncounts.wordpress.com/2012/10/25/mathematical-structures-5/ always busy counting, doubting every figured guess . . . Wed, 19 Jun 2013 18:15:17 +0000 hourly 1 http://wordpress.com/ By: Peter Cameron http://cameroncounts.wordpress.com/2012/10/25/mathematical-structures-5/#comment-8156 Mon, 29 Oct 2012 15:22:50 +0000 http://cameroncounts.wordpress.com/?p=2701#comment-8156 The students had actually seen Gauss’ method of adding the first n natural numbers on problem sheet 1: write them down in reverse order underneath and add by columns rather than rows. First years are not going to solve linear inhomogeneous recurrence relations.

I will say more about this in the next instalment.

]]> By: jan3grabowski http://cameroncounts.wordpress.com/2012/10/25/mathematical-structures-5/#comment-8148 Mon, 29 Oct 2012 12:49:43 +0000 http://cameroncounts.wordpress.com/?p=2701#comment-8148 What doesn’t help matters is that the use of induction can be circumvented in some cases, making it even more susceptible to scepticism. For example, one can prove the formula for the sum of the first n natural numbers using the methods of solving linear inhomogeneous recurrence relations. Even less helpfully, in that case the method produces the formula naturally – without having to guess it first. (There is still a “free” element of choice, in finding particular solutions, but it’s easier to guess what to do.)

The case of infinitely many primes is more convincing in my view: is there a “reasonable” alternative proof of that result avoiding induction, I wonder?

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