osman@sprite.DEC (Eric Osman, dtn 283-7484) (02/15/85)
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550 net.math... User unknown
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Date: Friday, 15 Feb 1985 05:58:31-PST
From: decwrl!dec-rhea!dec-sprite!osman (Eric Osman, dtn 283-7484)
To: harpo!whuxlm!whuxl!houxm!ihnp4!inuxc!pur-ee!uiucscsp!ashby@decvax,
decwrl!dec-rhea!dec-sprite!net.math
Subject: Ashby's solution is wrong (to 1+11+111 . . .+ n 1's)
> | Solution to Sn = 1 + 11 + 111 + ... + 11...11 |
> | |<--->| |
> | n 1's |
> | is easily obtained by re-writing as |
> | |
> | (1) Sn = (10^1 - 1)/9 + (10^2 - 1)/9 + ... + (10^n - 1)/9 |
> | |
> | re-arranging terms gives |
> | |
> | (2) Sn = 10*(10^n -1)/81 - n/9 |
> | |
> ===========================================================================
There's something wrong with this solution ! s4 = 1+11+111+1111 = 1234.
Let's check: s4 = 10*(10^4-1)/81 - 4/9 = 10*9999/81 - 4/9 = 1234 + 16/81 -
4/9 DOES NOT EQUAL 1234 !! [(4/9)^2 = 16/81 but so what]gjerawlins@watdaisy.UUCP (Gregory J.E. Rawlins) (02/17/85)
[ A lot of mail redirection stuff plus the original problem deleted] [ Problem is to find sum of series 1+11+111+...+ n 1's] [ Ashby's solution of the problem is....] >> (2) Sn = 10*(10^n -1)/81 - n/9 > >=========================================================================== > >There's something wrong with this solution ! s4 = 1+11+111+1111 = 1234. >Let's check: s4 = 10*(10^4-1)/81 - 4/9 = 10*9999/81 - 4/9 = 1234 + 16/81 - >4/9 DOES NOT EQUAL 1234 !! [(4/9)^2 = 16/81 but so what] ......perhaps you might check that 10*9999/81 = 1234 + 4/9 (not 16/81) -- Gregory Rawlins CS Dept.,U.Waterloo,Waterloo,Ont.N2L3G1 (519)884-3852 gjerawlins%watdaisy@waterloo.csnet CSNET gjerawlins%watdaisy%waterloo.csnet@csnet-relay.arpa ARPA {allegra|clyde|linus|inhp4|decvax}!watmath!watdaisy!gjerawlins UUCP
hopp@nbs-amrf.UUCP (Ted Hopp) (02/17/85)
From: osman@sprite.DEC (Eric Osman, dtn 283-7484) > Subject: Ashby's solution is wrong (to 1+11+111 . . .+ n 1's) > > | Solution to Sn = 1 + 11 + 111 + ... + 11...11 | > > | |<--->| | > > | n 1's | > > | is easily obtained by re-writing as | > > | | > > | (1) Sn = (10^1 - 1)/9 + (10^2 - 1)/9 + ... + (10^n - 1)/9 | > > | | > > | re-arranging terms gives | > > | | > > | (2) Sn = 10*(10^n -1)/81 - n/9 | > > | | > > =========================================================================== > There's something wrong with this solution ! s4 = 1+11+111+1111 = 1234. > Let's check: s4 = 10*(10^4-1)/81 - 4/9 = 10*9999/81 - 4/9 = 1234 + 16/81 - > 4/9 DOES NOT EQUAL 1234 !! [(4/9)^2 = 16/81 but so what] Check your math. 10*9999/81 = 1234 + 36/81, not 1234 + 16/81. -- Ted Hopp {seismo,umcp-cs}!nbs-amrf!hopp
chuck@dartvax.UUCP (Chuck Simmons) (02/18/85)
>> | Solution to Sn = 1 + 11 + 111 + ... + 11...11 | >> | |<--->| | >> | n 1's | >> | is easily obtained by re-writing as | >> | | >> | (1) Sn = (10^1 - 1)/9 + (10^2 - 1)/9 + ... + (10^n - 1)/9 | >> | | >> | re-arranging terms gives | >> | | >> | (2) Sn = 10*(10^n -1)/81 - n/9 | >> | | >> =========================================================================== > > There's something wrong with this solution ! s4 = 1+11+111+1111 = 1234. > Let's check: s4 = 10*(10^4-1)/81 - 4/9 = 10*9999/81 - 4/9 = 1234 + 16/81 - > 4/9 DOES NOT EQUAL 1234 !! [(4/9)^2 = 16/81 but so what] Oops! 10*9999/81-4/9 = (11110-4)/9 = 11106/9 = 1234 Ashby's solution is quite nice. We have: Sn = (10^1 - 1)/9 + (10^2 - 1)/9 + ... + (10^n - 1)/9 = 1/9 * ( (10^1-1) + (10^2-1) + ... + (10^n-1) ) = 1/9 * (sigma i=1..n of 10^i - n) = 1/9 * (10 * sigma i=0..n-1 of 10^i - n) = 1/9 * (10 * (10^n - 1)/9 - n) = 10*(10^n - 1)/ 81 - n/9 as desired.