Megapizza

Mega pizza will be divided among 100 people. First gets 1%, 2nd 2% of the remainder, 3rd 3% of the remainder, etc. Last 100th 100% of the remainder. Which person got the biggest portion?

Final Answer:

n =  10

Step-by-step explanation:

r1​=1001​⋅ 1=1001⋅ 1​=1001​=0.01 r2​=1002​⋅ 0.99=500099​=0.0198 r3​=1003​⋅ 0.9702≐0.0291 r4​=1004​⋅ 0.941094≐0.0376 r5​=1005​⋅ 0.90345024≐0.0452 r6​=1006​⋅ 0.858277728≐0.0515 r7​=1007​⋅ 0.80678106432≐0.0565 r8​=1008​⋅ 0.7503063898176≐0.06 r9​=1009​⋅ 0.69028187863219≐0.0621 r10​=10010​⋅ 0.62815650955529≐0.0628 r11​=10011​⋅ 0.56534085859977≐0.0622 r12​=10012​⋅ 0.50315336415379≐0.0604 r13​=10013​⋅ 0.44277496045534≐0.0576 r14​=10014​⋅ 0.38521421559614≐0.0539 r15​=10015​⋅ 0.33128422541268≐0.0497 r16​=10016​⋅ 0.28159159160078≐0.0451 r17​=10017​⋅ 0.23653693694466≐0.0402 r18​=10018​⋅ 0.19632565766406≐0.0353 r19​=10019​⋅ 0.16098703928453≐0.0306 r20​=10020​⋅ 0.13039950182047≐0.0261 r21​=10021​⋅ 0.10431960145638≐0.0219 r22​=10022​⋅ 0.082412485150538≐0.0181 r23​=10023​⋅ 0.06428173841742≐0.0148 r24​=10024​⋅ 0.049496938581413≐0.0119 r25​=10025​⋅ 0.037617673321874≐0.0094 r26​=10026​⋅ 0.028213254991405≐0.0073 r27​=10027​⋅ 0.02087780869364≐0.0056 r28​=10028​⋅ 0.015240800346357≐0.0043 r29​=10029​⋅ 0.010973376249377≐0.0032 r30​=10030​⋅ 0.0077910971370578≐0.0023 r31​=10031​⋅ 0.0054537679959405≐0.0017 r32​=10032​⋅ 0.0037630999171989≐0.0012 r33​=10033​⋅ 0.0025589079436953≐0.0008 r34​=10034​⋅ 0.0017144683222758≐0.0006 r35​=10035​⋅ 0.001131549092702≐0.0004 r36​=10036​⋅ 0.00073550691025633≐0.0003 r37​=10037​⋅ 0.00047072442256405≐0.0002 r38​=10038​⋅ 0.00029655638621535≐0.0001 r39​=10039​⋅ 0.00018386495945352≐0.0001 r40​=10040​⋅ 0.00011215762526665≐4.4863⋅10−5 r41​=10041​⋅ 6.7294575159988⋅10−5≐2.7591⋅10−5 r42​=10042​⋅ 3.9703799344393⋅10−5≐1.6676⋅10−5 r43​=10043​⋅ 2.3028203619748⋅10−5≐9.9021⋅10−6 r44​=10044​⋅ 1.3126076063256⋅10−5≐5.7755⋅10−6 r45​=10045​⋅ 7.3506025954235⋅10−6≐3.3078⋅10−6 r46​=10046​⋅ 4.0428314274829⋅10−6≐1.8597⋅10−6 r47​=10047​⋅ 2.1831289708408⋅10−6≐1.0261⋅10−6 r48​=10048​⋅ 1.1570583545456⋅10−6≐5.5539⋅10−7 r49​=10049​⋅ 6.0167034436372⋅10−7≐2.9482⋅10−7 r50​=10050​⋅ 3.068518756255⋅10−7≐1.5343⋅10−7 r51​=10051​⋅ 1.5342593781275⋅10−7=7.8247⋅10−8 r52​=10052​⋅ 7.5178709528247⋅10−8=3.9093⋅10−8 r53​=10053​⋅ 3.6085780573558⋅10−8=1.9125⋅10−8 r54​=10054​⋅ 1.6960316869572⋅10−8=9.1586⋅10−9 r55​=10055​⋅ 7.8017457600033⋅10−9=4.291⋅10−9 r56​=10056​⋅ 3.5107855920015⋅10−9=1.966⋅10−9 r57​=10057​⋅ 1.5447456604807⋅10−9=8.8051⋅10−10 r58​=10058​⋅ 6.6424063400668⋅10−10=3.8526⋅10−10 r59​=10059​⋅ 2.7898106628281⋅10−10=1.646⋅10−10 r60​=10060​⋅ 1.1438223717595⋅10−10=6.8629⋅10−11 r61​=10061​⋅ 4.575289487038⋅10−11=2.7909⋅10−11 r62​=10062​⋅ 1.7843628999448⋅10−11=1.1063⋅10−11 r63​=10063​⋅ 6.7805790197904⋅10−12=4.2718⋅10−12 r64​=10064​⋅ 2.5088142373224⋅10−12=1.6056⋅10−12 r65​=10065​⋅ 9.0317312543608⋅10−13=5.8706⋅10−13 r66​=10066​⋅ 3.1611059390263⋅10−13=2.0863⋅10−13 r67​=10067​⋅ 1.0747760192689⋅10−13=7.201⋅10−14 r68​=10068​⋅ 3.5467608635875⋅10−14=2.4118⋅10−14 r69​=10069​⋅ 1.134963476348⋅10−14=7.8312⋅10−15 r70​=10070​⋅ 3.5183867766788⋅10−15=2.4629⋅10−15 r71​=10071​⋅ 1.0555160330036⋅10−15=7.4942⋅10−16 r72​=10072​⋅ 3.0609964957105⋅10−16=2.2039⋅10−16 r73​=10073​⋅ 8.5707901879895⋅10−17=6.2567⋅10−17 r74​=10074​⋅ 2.3141133507572⋅10−17=1.7124⋅10−17 r75​=10075​⋅ 6.0166947119686⋅10−18=4.5125⋅10−18 r76​=10076​⋅ 1.5041736779922⋅10−18=1.1432⋅10−18 r77​=10077​⋅ 3.6100168271812⋅10−19=2.7797⋅10−19 r78​=10078​⋅ 8.3030387025167⋅10−20=6.4764⋅10−20 r79​=10079​⋅ 1.8266685145537⋅10−20=1.4431⋅10−20 r80​=10080​⋅ 3.8360038805627⋅10−21=3.0688⋅10−21 r81​=10081​⋅ 7.6720077611254⋅10−22=6.2143⋅10−22 r82​=10082​⋅ 1.4576814746138⋅10−22=1.1953⋅10−22 r83​=10083​⋅ 2.6238266543049⋅10−23=2.1778⋅10−23 r84​=10084​⋅ 4.4605053123183⋅10−24=3.7468⋅10−24 r85​=10085​⋅ 7.1368084997093⋅10−25=6.0663⋅10−25 r86​=10086​⋅ 1.0705212749564⋅10−25=9.2065⋅10−26 r87​=10087​⋅ 1.498729784939⋅10−26=1.3039⋅10−26 r88​=10088​⋅ 1.9483487204206⋅10−27=1.7145⋅10−27 r89​=10089​⋅ 2.3380184645048⋅10−28=2.0808⋅10−28 r90​=10090​⋅ 2.5718203109553⋅10−29=2.3146⋅10−29 r91​=10091​⋅ 2.5718203109553⋅10−30=2.3404⋅10−30 r92​=10092​⋅ 2.3146382798597⋅10−31=2.1295⋅10−31 r93​=10093​⋅ 1.8517106238878⋅10−32=1.7221⋅10−32 r94​=10094​⋅ 1.2961974367214⋅10−33=1.2184⋅10−33 r95​=10095​⋅ 7.7771846203287⋅10−35=7.3883⋅10−35 r96​=10096​⋅ 3.8885923101643⋅10−36=3.733⋅10−36 r97​=10097​⋅ 1.5554369240657⋅10−37=1.5088⋅10−37 r98​=10098​⋅ 4.6663107721972⋅10−39=4.573⋅10−39 r99​=10099​⋅ 9.3326215443944⋅10−41=9.2393⋅10−41 r100​=100100​⋅ 9.3326215443945⋅10−43=9.3326⋅10−43 r10​ > ∀r r10​ … max n=10

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Showing 1 comment:
Dr Math
Perhaps a more elegant solution: The Kth person gets 99! / (99 - (k-1)! * K / 100exp (k). K + 1. The person gets 99! / (99 - k)! * (K + 1) / 100exp (k + 1) The ratio m (k + 1) / m (k) must be greater than one, ie: the next person in the order will receive a larger portion than the previous one.Applies to k = 1, 2, … 9. The 10th person gets the largest portion.





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