Each of A and B is a positive real number.
Consider this function:
A*B
A o B = -----
A+B
Determine the value of:
21o(22o(23o ......o(22022 o 22023)))
Early on, with lower powers of two, the final (rightmost) power of two is the numerator of the answer, and the denominator is one less than the numerator:
final determined
exp't value
3 8/7
4 16/15
5 32/31
6 64/63
7 128/127
8 256/255
9 512/511
10 1024/1023
11 2048/2047
12 4096/4095
13 8192/8191
14 16384/16383
15 32768/32767
16 65536/65535
17 131072/131071
18 262144/262143
19 524288/524287
20 1048576/1048575
21 2097152/2097151
22 4194304/4194303
23 8388608/8388607
24 16777216/16777215
25 33554432/33554431
This continues through extremely large values of both numerator and denominator. At 2^2023 as the rightmost number, you get
9631218335423173696015738454064712512625486454282845268288357683278517
4664461287537804846201905350278880351665383273421210406896920475128576
4221918179043624419894139984279754512017898273159626328827668380262481
2208650177312678026009153751831792643806511654213677735639479033914667
6855708979226348173410849338514606325830049576416536529554633780885267
3629710735621386935094923561594142327134318905856137785813985574356271
6799186944470152944816918493419174323465595015026833030825915850745767
8696308503954644628109504872366923085654833908790992275376288406060765
9880382812905450025751549153093939827557015748608
/
9631218335423173696015738454064712512625486454282845268288357683278517
4664461287537804846201905350278880351665383273421210406896920475128576
4221918179043624419894139984279754512017898273159626328827668380262481
2208650177312678026009153751831792643806511654213677735639479033914667
6855708979226348173410849338514606325830049576416536529554633780885267
3629710735621386935094923561594142327134318905856137785813985574356271
6799186944470152944816918493419174323465595015026833030825915850745767
8696308503954644628109504872366923085654833908790992275376288406060765
9880382812905450025751549153093939827557015748607
The numerator and denominator are each 609 digits long with the latter 1 unit less, so it's quite close to 1.
prod=sym(2)^2023;
for i=2022:-1:1
prod=prod*sym(2)^i/(prod+sym(2)^i)
pr=char(prod)
disp([i,length(pr),strfind(pr,'/')])
end
disp(prod)
disp([i,length(pr),strfind(pr,'/')])
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Posted by Charlie
on 2023-08-08 08:46:00 |
computer solution
