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Evaluate this sum (Posted on 2005-12-21) Difficulty: 4 of 5
Evaluate this sum in terms of n (the number of terms):
        1      2       4        8          16         
   S = --- + ----- + ----- + ------- + ---------- + ..... 
        2      5       41     3,281    21,523,361              

See The Solution Submitted by pcbouhid    
Rating: 4.3333 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Using Charlie's idea (spoiler?) -- numerical verification | Comment 5 of 14 |
(In reply to re: Using Charlie's idea (spoiler?) by Charlie)

Using UBASIC, we can varify this formula for the first ten terms:

   10   Num=1:Den=2:N=0
   15   for G=1 to 10
   20    Tot=Tot+Num//Den
   30    print 1-Tot
   35    print 2^(N+2)//(3^(2^(N+1))-1)
   40    Num=Num*2:N=N+1
   50    X=Num//(1-Tot)
   60    Den=X+1
   70   next

produces

 1//2
 1//2
 1//10
 1//10
 1//410
 1//410
 1//1345210
 1//1345210
 1//28953440450810
 1//28953440450810
 1//26825654846035253786389446010
 1//26825654846035253786389446010
 1//46055408506791340513753409614892651037805514032327504332410
 1//46055408506791340513753409614892651037805514032327504332410
 1//2715008835491108061815230210735572326449586929854084726398531727149142043550
12570197269080379120331097288409258884569210
 1//2715008835491108061815230210735572326449586929854084726398531727149142043550
12570197269080379120331097288409258884569210
 1//1887045882059464365766998898634724712441549390781575053728958752822506518204
64228515800189744004192913965647747017442573292514737317704500821269926550084370
53281787239968163557400451486725949548946331444988517848234171391268540901362570
738810
 1//1887045882059464365766998898634724712441549390781575053728958752822506518204
64228515800189744004192913965647747017442573292514737317704500821269926550084370
53281787239968163557400451486725949548946331444988517848234171391268540901362570
738810
 1//1823202386430761931313280049730793887819383675701826209719906893174214065513
99048452145355927150379502182014192105212479545276882125078544229175457511137298
49325768974329227584902686521091621303006436427785482484804560689722542383415069
20402327025056734142513296124194950853984661086856858665984084501641613166737341
02895792062757353021853638404696188190113350547238585462655629485074639831902863
27416275365059420227725584613623384131538049634975449146860290491904115078665073
3561382010
 1//1823202386430761931313280049730793887819383675701826209719906893174214065513
99048452145355927150379502182014192105212479545276882125078544229175457511137298
49325768974329227584902686521091621303006436427785482484804560689722542383415069
20402327025056734142513296124194950853984661086856858665984084501641613166737341
02895792062757353021853638404696188190113350547238585462655629485074639831902863
27416275365059420227725584613623384131538049634975449146860290491904115078665073
3561382010

showing a match between 1 minus the sum, and the formula

2^(N+2)/(3^(2^(N+1))-1)

for n = 0 to 9


  Posted by Charlie on 2005-12-21 16:18:45
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