A Combi Problem by Michael Huang

The numbering can be done in different ways, so the question is somewhat ambiguous, but I use the following: definition for the summands in Newton's binomium $(a+b)^n=\sum_{i=0}^n { {n\choose i} a^{n-i} b^i}$ .

So the $i^{th}$ summand is $s_i={n\choose i} a^{n-i} b^i=\frac{n!}{i!(n-i)!} a^{n-i} b^i$

The ratio of two subsequent summands is $\frac{s_i}{s_{i-1}} = \frac{\frac{n!}{i!(n-i)!} a^{n-i} b^i}{\frac{n!}{(i-1)!(n-i+1)!} a^{n-i+1} b^{i-1} }=\frac{(n-i+1)b}{ia}$ .

If this quantity exceeds 1, then that means that $s_i \gt s_{i-1}$ and we find the maximum by setting $(n-i+1)b=ia$ or $i=\frac{(n+1)b}{a+b}=\frac{101 \sqrt{2}}{1+\sqrt{2}}=59.16...$

So the i for which the value is the largest it is either 59 or 60. Checking $\frac{s_{60}}{s_{59}} =\frac{41\sqrt{2}}{60}\lt 1$ shows that $s_{59}$ is the largest so that maximum is at $i=\boxed{59}$ .

$t=\text{List}\text{@@}\text{Expand}\left[(a+b)^{100}\right]\text{/.}\, \left\{a\to 1,b\to \sqrt{2}\right\}; \text{Position}[t,\max (t)][[1,1]]-1 \Rightarrow 59$

The largest magnitude summand is $10799931603545409340755368688903782400 \sqrt{2}$ . The list of values is

$\begin{array}{rr} 1 & 0 \\ 100 \sqrt{2} & 1 \\ 9900 & 2 \\ 323400 \sqrt{2} & 3 \\ 15684900 & 4 \\ 301150080 \sqrt{2} & 5 \\ 9536419200 & 6 \\ 128060486400 \sqrt{2} & 7 \\ 2977406308800 & 8 \\ 30435708934400 \sqrt{2} & 9 \\ 553929902606080 & 10 \\ 4532153748595200 \sqrt{2} & 11 \\ 67226947270828800 & 12 \\ 455074719987148800 \sqrt{2} & 13 \\ 5655928662697420800 & 14 \\ 32427324332798545920 \sqrt{2} & 15 \\ 344540321035984550400 & 16 \\ 1702434527471923660800 \sqrt{2} & 17 \\ 15700229531129962649600 & 18 \\ 67758885344876680908800 \sqrt{2} & 19 \\ 548846971293501115361280 & 20 \\ 2090845604927623296614400 \sqrt{2} & 21 \\ 15016072980843840039321600 & 22 \\ 50924073587209544481177600 \sqrt{2} & 23 \\ 326762805517927910420889600 & 24 \\ 993358928774500847679504384 \sqrt{2} & 25 \\ 5730916896775966428920217600 & 26 \\ 15706957420793389471855411200 \sqrt{2} & 27 \\ 81900563694136959388960358400 & 28 \\ 203339330550960726758798131200 \sqrt{2} & 29 \\ 962472831274547439991644487680 & 30 \\ 2173325748039300670948874649600 \sqrt{2} & 31 \\ 9372467288419484143467021926400 & 32 \\ 19312962897349240053204772454400 \sqrt{2} & 33 \\ 76115794948376416680277632614400 & 34 \\ 143532641902652671454237821501440 \sqrt{2} & 35 \\ 518312317981801313584747688755200 & 36 \\ 896540225698250920795239245414400 \sqrt{2} & 37 \\ 2972738643104726737373688024268800 & 38 \\ 4725892201858796351722273269350400 \sqrt{2} & 39 \\ 14413971215669328872752933471518720 & 40 \\ 21093616413174627618662829470515200 \sqrt{2} & 41 \\ 59263017541776334738147949464780800 & 42 \\ 79936163195884358484013513231564800 \sqrt{2} & 43 \\ 207107331916609474254035011554508800 & 44 \\ 257733568607336234627243569934499840 \sqrt{2} & 45 \\ 616319403191456213239060710712934400 & 46 \\ 708111654730609266274665497414860800 \sqrt{2} & 47 \\ 1563746570863428796356552973457817600 & 48 \\ 1659486156834659130827362339179724800 \sqrt{2} & 49 \\ 3385351759942704626887819171926638592 & 50 \\ 3318972313669318261654724678359449600 \sqrt{2} & 51 \\ 6254986283453715185426211893831270400 & 52 \\ 5664893237844874130197323979318886400 \sqrt{2} & 53 \\ 9861110451063299411824971371406950400 & 54 \\ 8247474195434759508071794237903994880 \sqrt{2} & 55 \\ 13254869242663006352258240739488563200 & 56 \\ 10231828889073197885953729693640294400 \sqrt{2} & 57 \\ 15171332490694741692965875062983884800 & 58 \\ 10799931603545409340755368688903782400 \sqrt{2} & 59 \\ 14759906524845392765699003874835169280 & 60 \\ 9678627229406814928327215655629619200 \sqrt{2} & 61 \\ 12176337482156960716282626147405004800 & 62 \\ 7344457528920071543154599898434764800 \sqrt{2} & 63 \\ 8492029017813832721772506132565196800 & 64 \\ 4703277609866122738212464934959185920 \sqrt{2} & 65 \\ 4988324737736796843558674931017318400 & 66 \\ 2531388672881359592253655935143116800 \sqrt{2} & 67 \\ 2456936064855437251305018995874201600 & 68 \\ 1139448609788028870170443592289484800 \sqrt{2} & 69 \\ 1009225911526539856436678610313543680 & 70 \\ 426433483743608390043667018442342400 \sqrt{2} & 71 \\ 343515861904573425312953987078553600 & 72 \\ 131759508675726793270722077235609600 \sqrt{2} & 73 \\ 96148830655260092386743137442201600 & 74 \\ 33331594627156832027404287646629888 \sqrt{2} & 75 \\ 21928680675761073702239662925414400 & 76 \\ 6834913457380074920178596236492800 \sqrt{2} & 77 \\ 4030846397942095465746351626649600 & 78 \\ 1122514186768684813245819440332800 \sqrt{2} & 79 \\ 589319948053559526954055206174720 & 80 \\ 145511098284829512828161779302400 \sqrt{2} & 81 \\ 67431972375896603505733507481600 & 82 \\ 14623801238146251362689194393600 \sqrt{2} & 83 \\ 5919157644011577932517054873600 & 84 \\ 1114194380049238199062033858560 \sqrt{2} & 85 \\ 388672458156710999672802508800 & 86 \\ 62544993266597172361140633600 \sqrt{2} & 87 \\ 18479202556040073652155187200 & 88 \\ 2491577872724504312650137600 \sqrt{2} & 89 \\ 609052368888212165314478080 & 90 \\ 66928831745957380803788800 \sqrt{2} & 91 \\ 13094771428556878852915200 & 92 \\ 1126431950843602481971200 \sqrt{2} & 93 \\ 167766460763940795187200 & 94 \\ 10595776469301523906560 \sqrt{2} & 95 \\ 1103726715552242073600 & 96 \\ 45514503734113075200 \sqrt{2} & 97 \\ 2786602269435494400 & 98 \\ 56294995342131200 \sqrt{2} & 99 \\ 1125899906842624 & 100 \\ \end{array}$

$(1+\sqrt{2})^{100} = \frac{{{100}\choose{i}}(\sqrt{2})}{{{100}\choose{i-1}}(\sqrt{2})^{i-1}}$ = $\frac{(101-i)!(i-1)!}{100}$ = $\frac{100!}{(100-i)i!}$ = $\frac{101-i}{i}\sqrt{2}>i$ = $101\sqrt{2}-i\sqrt{2}>i$ = $\frac{i(1+\sqrt{2})}{1+\sqrt{2}}<\frac{101\sqrt{2}}{1+\sqrt{2}}\cdot\frac{1-\sqrt{2}}{1-\sqrt{2}}$ = $-101\sqrt{2}+202\approx{59.16}$ $\boxed{i=59}$