Re: Re: Re: Print[Plot] vs Print[text,Plot]? (*now Do and Table*)

*To*: mathgroup at smc.vnet.net*Subject*: [mg88176] Re: [mg88113] Re: [mg88090] Re: Print[Plot] vs Print[text,Plot]? (*now Do and Table*)*From*: DrMajorBob <drmajorbob at att.net>*Date*: Sun, 27 Apr 2008 04:58:49 -0400 (EDT)*References*: <200804240957.FAA28500@smc.vnet.net> <4714471.1209200633216.JavaMail.root@m08>*Reply-to*: drmajorbob at longhorns.com

Your curious result was Clear[n, result] result = Sum[i (i + 1)/2, {i, 1, 10^7}] 166666716666670000000 In general, the sum is Clear[k, result, n] result = Sum[i (i + 1)/2, {i, 1, n}] 1/6 n (1 + n) (2 + n) If n is the kth power of 10, the factor n causes "result" to have k trailing zeroes, and the factor (n+1) causes the repetition you noticed, but it also requires (n+2)/6 to be an integer, which occurs when Mod[10^k+2,6]==0. But, modulo 6, we see that 10 is congruent to 4, and 4 is idempotent: Mod[10, 6] 4 Mod[4^2, 6] 4 Union@Table[Mod[10^k, 6], {k, 100}] {4} As a consequence, the pattern you noticed will always appear when n is a power of 10. Table[Mod[10^k, 6], {k, 100}] {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, \ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, \ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, \ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4} Oops! I almost forgot the final requirement, that (n+2)/6 < n. But that's pretty obvious. Bobby On Fri, 25 Apr 2008 04:28:10 -0500, Syd Geraghty <sydgeraghty at mac.com> wrote: > Craig, > > Just a (tongue in cheek) reminder that Mathematica is a wonderful all > purpose environment in which to do Mathematics. > > 221 years ago the 10 year old Gauss would have solved the problem by > coding:- > > n = 10000000; > > Timing[total = n (n + 1)/2] > > {0.000035, 50000005000000} > > Which is a whole lot faster! > > I realize that this is not very helpful in deciding usage of Table vs > Do. > > But the precocious Gauss also might have tried > > Timing[total = Table[i (i + 1)/2, {i, 10000000}]] > > {20.46425400000001, {1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, > 91, <<9999974>>, > 49999885000066, 49999895000055, 49999905000045, 49999915000036, > 49999925000028, 49999935000021, 49999945000015, 49999955000010, > 49999965000006, 49999975000003, 49999985000001, 49999995000000, > 50000005000000}} > > or better yet > > Timing[Total[Table[i (i + 1)/2, {i, 10000000}]]] > > {31.7433, 166666716666670000000} > > Just to show off ... :) > > Cheers ... Syd > > PS Where would we be now if Stephen Wolfram had provided Gauss with a > free Mathematica License back then? > > BTW What a surprising result (1666667)(1666667)(0000000) > > I wonder if anyone has computed this result before and explained its > repetition of the first 7 digits? > > > Syd Geraghty B.Sc, M.Sc. > > sydgeraghty at mac.com > > My System > > Mathematica 6.0.2.1 for Mac OS X x86 (64 - bit) (March 13, 2008) > MacOS X V 10.5.2 > MacBook Pro 2.33 Ghz Intel Core 2 Duo 2GB RAM > > > > > > > On Apr 24, 2008, at 2:57 AM, W_Craig Carter wrote: > >> This is problem with a known simple result, but it will serve: let's >> find the sum of the first 10000000 integers. >> >> (*let's use a do loop*) >> Timing[ >> icount = 0; >> Do[icount = icount + i, {i, 1, 10000000, 1}]; >> icount >> ] >> >> (*this returns {10.2826, 50000005000000} on my machine.*) >> (*10.28 being a measure of how long the computation took to run*) >> >> (*lets try a table*) >> Timing[ >> Total[Table[i, {i, 1, 10000000, 1}]] >> ] >> >> (*This returns {3.25516, 50000005000000} on my machine*) > > > -- DrMajorBob at longhorns.com

**References**:**Re: Print[Plot] vs Print[text,Plot]? (*now Do and Table*)***From:*"W_Craig Carter" <ccarter@mit.edu>