The Value
110+210 +310 +410.... +100010 = ?
Believe it or not famous Mathematician Jakob Bernoulli determined the value in less than 10 minutes.
How long will you take ?
Believe it or not famous Mathematician Jakob Bernoulli determined the value in less than 10 minutes.
How long will you take ?
Labels: friday special, mathschallenge
posted by Rajesh Lal at
2:35 PM
18 Comments:
91409924241424243424241924242500
However long it takes me to type it into my calculator.
If you 10th root each term, then you get 1+2+...+1000=10th root of "?".
Now, using the "Gaos(?)" method, 1+2+3....+1000=1000(1000+1)/2=1000*1001/2=500*1001=500500.
Now the "?" is the answer and 1+2+3...+1000 is only the 10th root of "?" which is 500500. Now so the real value of "?" is 500500 to the tenth power, which would be 98637218770532442394535645507910000000000000000000000000 which is a very large amount.
And that took me around 5 minutes with a caculator to do 1 task - 500500^10.
0.045s real time.
91409924241424243424241924242500
is probably more accurate than previous computations.
-h
oops... except Anons first post - sorry - :-)
-h
959924142434242000000
I just used an excel spreadsheet. :)
This may introduce error, though.
James Yu said...
If you 10th root each term, then you get 1+2+...+1000=10th root of "?".
This is false. (1+2+3....+1000)^10 is not equal to 1^10 + 2^10 +3^10...+1000^10
The guy who worked out a 21 digit answer using an Excel spreadsheet is wrong. The last term alone has 31 digits (a 1 followed by 30 0s).
See: http://numbers.computation.free.fr/Constants/Miscellaneous/bernoulli.html
for the general formulae and the answer to this question.
I got the same answer as the first poster. But it took several days.
91,409,924,241,424,243,424,241,924,242,500
Below are three numbers per line
i.............I^10...........sum(i^10)
{1,1,1}
{2,1024,1025}
{3,59049,60074}
{4,1048576,1108650}
{5,9765625,10874275}
{6,60466176,71340451}
{7,282475249,353815700}
{8,1073741824,1427557524}
{9,3486784401,4914341925}
{10,10000000000,14914341925}
{11,25937424601,40851766526}
{12,61917364224,102769130750}
{13,137858491849,240627622599}
{14,289254654976,529882277575}
{15,576650390625,1106532668200}
{16,1099511627776,2206044295976}
{17,2015993900449,4222038196425}
{18,3570467226624,7792505423049}
{19,6131066257801,13923571680850}
{20,10240000000000,24163571680850}
{21,16679880978201,40843452659051}
{22,26559922791424,67403375450475}
{23,41426511213649,108829886664124}
{24,63403380965376,172233267629500}
{25,95367431640625,267600699270125}
{26,141167095653376,408767794923501}
{27,205891132094649,614658927018150}
{28,296196766695424,910855693713574}
{29,420707233300201,1331562927013775}
{30,590490000000000,1922052927013775}
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{33,1531578985264449,5399160106101649}
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{995,951110130465771892558603515625,86508586167410818764988816533850}
{996,960712373502810196866499608576,87469298540913628961855316142426}
{997,970401776948916827855048229049,88439700317862545789710364371475}
{998,980179043351949424651514881024,89419879361214495214361879252499}
{999,990044880209748209880044990001,90409924241424243424241924242500}
{1000,1000000000000000000000000000000,91409924241424243424241924242500}
with the distributive property,
can be rewritten to
(1+2+3..+1000) ^ 10
((1000*1001)/2)^10
(1001000/2) ^10
(500500)^10
9.863721877053244239453564550791e+56
Only used the calculator on the last step..
oh.. and that took me about 3 minutes
I think you missed it by 25 decimal places.
To the Anon who spoke of the distributive property....
There's something wrong with your procedure. I am not sure what the problem is, but when I check your method it goes wrong from the beginning.
Suppose the problem was just up to ten, not 1000.
At ten, your method gives 253,295,162,119,141,000
when it should have been
14,914,341,925.
You can check the numbers up to ten
in the long, long list above.
I would appreciate your checking this out and letting us know what
was wrong with your procedure.
500500^10
The sum of y^10 from y=x-999 to y=x is equal to 1000x^10-4995000x^9+14977507500x^8-29940030000000x^7+41895069999993000x^6-41874104999979000000x^5+29895104999965000005000x^4-14940069999965000010000000x^3+4977529999979000009999998500x^2-995007499993000004999998500000x+90409924241424243424241924242500. Pluggin in 1000 for x yields 91409924241424243424241924242500.
500500^10=
986372187705324423945356455079101562500000000000000000000
Which is the wrong answer.
The sum is
91409924241424243424241924242500
More generally sum(k^10,k=1 to n)=
(6*n^11+33*n^10+55*n^9-66*n^7+66*n^5-33*n^3+5*n)/66
Jakob Bernoulli showed this using his Bernoulli coefficients.
I used wolframalpha:
http://www.wolframalpha.com/input/?i=Sum%5Bk%5E10%2C+%7Bk%2C+1%2C+1000%7D%5D
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