Note Young Gauss Meets Dynamical Systems CONSTANTIN P. NICULESCU M M ost people are convinced that doing mathematics is something like computing sums such as S ¼ 1 þ 2 þ 3 þþ 100: But we know that one who does this by merely add- ing terms one after another is not seeing the forest for the trees. An anecdote about young Gauss tells us that he solved the above problem by noticing that pairwise addition of terms from opposite ends of the list yields identical inter- mediate sums. This famous story is well told by Hayes in [5], with references. A very convenient way to express Gauss’s idea is to write down the series twice, once in ascending and once in descending order, 1 þ 2 þ 3 þ  þ 100 100 þ 99 þ 98 þ  þ 1 and to sum columns before summing rows. Thus 2S ¼ð1 þ 100Þþð2 þ 99Þþþð100 þ 1Þ ¼ 101 þ 101 þþ 101 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 100 times ¼ 10100; whence S ¼ 5050: Of course, the same technique applies to any arithmetic progression a 1 ; a 2 ¼ a 1 þ r ; a 3 ¼ a 1 þ 2r ; ...; a n ¼ a 1 þðn  1Þr ; ð1Þ and the result is the well-known summation formula a 1 þ a 2 þþ a n ¼ nða 1 þ a n Þ 2 : ð2Þ A similar idea can be used to sum up strings that are not necessarily arithmetic progressions. For example, n 0   a 0 þ n 1   a 1 þþ n n   a n ¼ 2 n1 ða 0 þ a n Þ; for every arithmetic progression a 0 ; a 1 ; ...; a n : Seventy years ago, A. L. O’Toole [11] recommended that teachers avoid the above derivation of the formula (2), considering it a mere trick that offers no insight. Instead, he called attention to the fundamental theorem of summation, a discrete variant of the Leibniz-Newton theorem: If there is a function f(x) such that a k = f(k + 1) - f(k) for k 2 f1; ...; ng; then X n k¼1 a k ¼ f ðn þ 1Þ f ð1Þ¼ f ðkÞj nþ1 1 : Indeed, this theorem provides a unifying approach for many interesting summation formulae (including those for arithmetic progressions and geometric progressions). However, determining the nature of the function f(x) is not always immediate. In the case of an arithmetic progression (1) we may choose f(x) as a second-degree polynomial, namely, f ðxÞ¼ r 2 x 2 þða 1  3r 2 Þx þ C ; where C is an arbitrary constant. Though more limited, ‘‘Gauss’s trick’’ is much simpler, and besides, it provides a nice illustration of a key concept of contemporary mathematics, that of measurable dynami- cal system. Letting M ¼f1; ...; ng; we may consider the measurable space ðM; PðMÞ; lÞ; where P(M) is the power set of M and l is the counting measure on M, defined by the formula lðAÞ¼ A j j for every A 2 PðMÞ: Every real sequence a 1 ; ...; a n of length n can be thought of as a function f : M ! R; given by f(k) = a k . Moreover, f is integrable with respect to l, and Z M f ðkÞdl ¼ a 1 þþ a n : The main ingredient that makes possible an easy compu- tation of the sum of an arithmetic progression is the existence of a nicely behaved map, namely, T : M ! M; T ðkÞ¼ n  k þ 1: Indeed, the measure l is invariant under the map T in the sense that Z M f ðkÞdl ¼ Z M f ðT ðkÞÞdl ð3Þ regardless of the choice of f (for T is just a permutation of the summation indices). When f represents an arithmetic progression of length n, then there exists a positive constant C such that Ó 2011 Springer Science+Business Media, LLC