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Find-how-many-distinct-integers-are-there-in-this-sequence-1-2-1-100-2-2-2-100-3-2-3-100-100-2-100-100-where-x-is-the-greatest-integer-that-is-less-than-




Question Number 176155 by adhigenz last updated on 14/Sep/22
Find how many distinct integers are there in this sequence:  ⌊((1^2 +1)/(100))⌋, ⌊((2^2 +2)/(100))⌋, ⌊((3^2 +3)/(100))⌋, ..., ⌊((100^2 +100)/(100))⌋  where ⌊x⌋ is the greatest integer that is less than or equal to x
$$\mathrm{Find}\:\mathrm{how}\:\mathrm{many}\:\mathrm{distinct}\:\mathrm{integers}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\mathrm{this}\:\mathrm{sequence}: \\ $$$$\lfloor\frac{\mathrm{1}^{\mathrm{2}} +\mathrm{1}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{2}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{3}}{\mathrm{100}}\rfloor,\:…,\:\lfloor\frac{\mathrm{100}^{\mathrm{2}} +\mathrm{100}}{\mathrm{100}}\rfloor \\ $$$$\mathrm{where}\:\lfloor{x}\rfloor\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{that}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:{x} \\ $$
Commented by Rasheed.Sindhi last updated on 14/Sep/22
102?
$$\mathrm{102}? \\ $$
Commented by mr W last updated on 14/Sep/22
there are totally 100 integers. so the  number of distinct integers must be  less than 100.
$${there}\:{are}\:{totally}\:\mathrm{100}\:{integers}.\:{so}\:{the} \\ $$$${number}\:{of}\:{distinct}\:{integers}\:{must}\:{be} \\ $$$${less}\:{than}\:\mathrm{100}. \\ $$
Commented by Rasheed.Sindhi last updated on 14/Sep/22
You′re very right sir!I′ve committed  mistake.
$${You}'{re}\:{very}\:{right}\:{sir}!{I}'{ve}\:{committed} \\ $$$${mistake}. \\ $$
Answered by Rasheed.Sindhi last updated on 14/Sep/22
⌊((1^2 +1)/(100))⌋, ⌊((2^2 +2)/(100))⌋, ⌊((3^2 +3)/(100))⌋, ..., ⌊((100^2 +100)/(100))⌋    ⌊((n^2 +n)/(100))⌋=⌊((  ((n(n+1))/2)  )/((100)/2))⌋=⌊((  ((n(n+1))/2)  )/(50))⌋      0≤((n(n+1))/2)<50 ;n=1,2,...,9:        ⌊((  ((n(n+1))/2)  )/(50))⌋=0  50≤((n(n+1))/2)<100;n=10,11,...,13:        ⌊((  ((n(n+1))/2)  )/(50))⌋=1  ....  ...  50k≤((n(n+1))/2)<50(k+1):        ⌊((  ((n(n+1))/2)  )/(50))⌋=k  We′ve to solve many such   inequalities...  Continue...
$$\lfloor\frac{\mathrm{1}^{\mathrm{2}} +\mathrm{1}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{2}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{3}}{\mathrm{100}}\rfloor,\:…,\:\lfloor\frac{\mathrm{100}^{\mathrm{2}} +\mathrm{100}}{\mathrm{100}}\rfloor \\ $$$$ \\ $$$$\lfloor\frac{{n}^{\mathrm{2}} +{n}}{\mathrm{100}}\rfloor=\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\frac{\mathrm{100}}{\mathrm{2}}}\rfloor=\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\mathrm{50}}\rfloor \\ $$$$\:\: \\ $$$$\mathrm{0}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}<\mathrm{50}\:;{n}=\mathrm{1},\mathrm{2},…,\mathrm{9}: \\ $$$$\:\:\:\:\:\:\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\mathrm{50}}\rfloor=\mathrm{0} \\ $$$$\mathrm{50}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}<\mathrm{100};{n}=\mathrm{10},\mathrm{11},…,\mathrm{13}: \\ $$$$\:\:\:\:\:\:\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\mathrm{50}}\rfloor=\mathrm{1} \\ $$$$…. \\ $$$$… \\ $$$$\mathrm{50}{k}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}<\mathrm{50}\left({k}+\mathrm{1}\right): \\ $$$$\:\:\:\:\:\:\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\mathrm{50}}\rfloor={k} \\ $$$${We}'{ve}\:{to}\:{solve}\:{many}\:{such} \\ $$$$\:{inequalities}… \\ $$$${Continue}… \\ $$
Answered by Rasheed.Sindhi last updated on 16/Sep/22
≪“SOMETHING is better than NOTHING”_(■ A       N       S       W       E       R ■) ≫  ⌊((1^2 +1)/(100))⌋, ⌊((2^2 +2)/(100))⌋, ⌊((3^2 +3)/(100))⌋, ..., ⌊((100^2 +100)/(100))⌋    ⌊((n^2 +n)/(100))⌋=⌊((  ((n(n+1))/2)  )/((100)/2))⌋=⌊((  ((n(n+1))/2)  )/(50))⌋      50k≤((n(n+1))/2)<50(k+1) ,k∈{0,1,2,...}:        ⌊((  ((n(n+1))/2)  )/(50))⌋=k  For (N/(D∈N)) to be integer, N should be  integral multiple of D   determinant ((n,(n(n+1)/2_(cumulative total^(OR) ) )),(1,1),(2,3),(3,6),(4,(10)),(5,(15)),(6,(21)),(7,(28)),(8,(36)),(9,(45<50   (1))),((10),(55)),((11),(66)),((12),(78)),((13),(91<100   (2))),((14),(105)),((15),(120)),((16),(136<150 (3))),((17),(153)),((18),(171)),((19),(190<200 (4))),((20),(210)),((21),(231<250 (5))),((22),(253)),((23),(276<300 (6))),((24),(300)),((25),(325<350 (7))),((26),(351)),((27),(378<400 (8))),((28),(406)),((29),(435<450 (9))),((30),(465)),((31),(496<500 (10))),((32),(528<550 (11))),((33),(561)),((34),(595<600 (12))),((35),(630<650 (13))),((36),(666<700 (14))),((37),(703)),((38),(741<750 (15))),((39),(780<800 (16)),((40),(820<850 (17))),((41),(861<900 (18))),((42),(903)),((43),(946<950 (19))),((44),(990<1000 (20))),((45),(1035<1050 (21))),((46),(1081<1100 (22))),((47),(1128<1150 (23))),((48),(1176<1200 (24))),((49),(1225<1250 (25))),((50),(1275<1300 (26))))  determinant ((n,(n(n+1)/2_(cumulative total^(OR) ) )),((51),(1326<1350 (27))),((52),(1378<1400 (28))),((53),(1431<1450 (29))),((54),(1485<1500 (30)),((55),(1540<1550 (31))),((56),(1596<1600 (32))),((57),(1653<1700 (33))),((58),(1711<1750 (34))),((59),(1770<1800 (35))),((60),(1830<1850 (36))),((61),(1891<1900 (37))),((62),(1953<2000 (38))),((63),(2016<2050 (39))),((64),(2080<2100 (40))),((65),(2145<2150 (41))),((66),(2211<2250 (42))),((67),(2278<2300 (43))),((68),(2346<2350 (44))),((69),(2415<2450 (45))),((70),(2485<2500 (46))),((71),(2556<2600 (47))),((72),(2628<2650 (48))),((73),(2701<2750 (49))),((74),(2775<2800 (50))),((75),(2850<2900 (51))),((76),(2926<2950 (52))),((77),(3003<3050 (53))),((78),(3081<3100 (54))),((79),(3160<3200 (55))),((80),(3240<3250 (56))),((81),(3321<3350 (57))),((82),(3403<3450 (58))),((83),(3486<3500 (59))),((84),(3570<3600 (60))),((85),(3655<3700 (61))),((86),(3741<3750 (62))),((87),(3828<3850 (63))),((88),(3916<3950 (64))),((89),(4005<4050 (65))),((90),(4095<4100 (66))),((91),(4186<4200 (67))),((92),(4278<4300 (68))),((93),(4371<4400 (69))),((94),(4465<4500 (70))),((95),(4560<4600 (71))),((96),(4656<4700 (72))),((97),(4753<4800 (73))),((98),(4851<4900 (74))),((99),(4950<5000 (75))),((100),(5050<5100 (76)_( determinant (((     76     )))) )))  Changing colour  of font here   means ′other distinct integer′.
$$\ll\underset{\blacksquare\:\boldsymbol{\mathrm{A}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{N}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{S}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{W}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{E}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{R}}\:\blacksquare} {“\boldsymbol{\mathrm{SOMETHING}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{better}}\:\boldsymbol{\mathrm{than}}\:\boldsymbol{\mathrm{NOTHING}}''}\gg \\ $$$$\lfloor\frac{\mathrm{1}^{\mathrm{2}} +\mathrm{1}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{2}^{\mathrm{2}} +\mathrm{2}}{\mathrm{100}}\rfloor,\:\lfloor\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{3}}{\mathrm{100}}\rfloor,\:…,\:\lfloor\frac{\mathrm{100}^{\mathrm{2}} +\mathrm{100}}{\mathrm{100}}\rfloor \\ $$$$ \\ $$$$\lfloor\frac{{n}^{\mathrm{2}} +{n}}{\mathrm{100}}\rfloor=\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\frac{\mathrm{100}}{\mathrm{2}}}\rfloor=\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\mathrm{50}}\rfloor \\ $$$$\:\: \\ $$$$\mathrm{50}{k}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}<\mathrm{50}\left({k}+\mathrm{1}\right)\:,{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…\right\}: \\ $$$$\:\:\:\:\:\:\lfloor\frac{\:\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:\:}{\mathrm{50}}\rfloor={k} \\ $$$${For}\:\frac{{N}}{{D}\in\mathbb{N}}\:{to}\:{be}\:{integer},\:{N}\:{should}\:{be} \\ $$$${integral}\:{multiple}\:{of}\:{D} \\ $$$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}{{n}}&\hline{\underset{\overset{\mathrm{OR}} {{cumulative}\:{total}}} {{n}\left({n}+\mathrm{1}\right)/\mathrm{2}}}\\{\mathrm{1}}&\hline{\mathrm{1}}\\{\mathrm{2}}&\hline{\mathrm{3}}\\{\mathrm{3}}&\hline{\mathrm{6}}\\{\mathrm{4}}&\hline{\mathrm{10}}\\{\mathrm{5}}&\hline{\mathrm{15}}\\{\mathrm{6}}&\hline{\mathrm{21}}\\{\mathrm{7}}&\hline{\mathrm{28}}\\{\mathrm{8}}&\hline{\mathrm{36}}\\{\mathrm{9}}&\hline{\mathrm{45}<\mathrm{50}\:\:\:\left(\mathrm{1}\right)}\\{\mathrm{10}}&\hline{\mathrm{55}}\\{\mathrm{11}}&\hline{\mathrm{66}}\\{\mathrm{12}}&\hline{\mathrm{78}}\\{\mathrm{13}}&\hline{\mathrm{91}<\mathrm{100}\:\:\:\left(\mathrm{2}\right)}\\{\mathrm{14}}&\hline{\mathrm{105}}\\{\mathrm{15}}&\hline{\mathrm{120}}\\{\mathrm{16}}&\hline{\mathrm{136}<\mathrm{150}\:\left(\mathrm{3}\right)}\\{\mathrm{17}}&\hline{\mathrm{153}}\\{\mathrm{18}}&\hline{\mathrm{171}}\\{\mathrm{19}}&\hline{\mathrm{190}<\mathrm{200}\:\left(\mathrm{4}\right)}\\{\mathrm{20}}&\hline{\mathrm{210}}\\{\mathrm{21}}&\hline{\mathrm{231}<\mathrm{250}\:\left(\mathrm{5}\right)}\\{\mathrm{22}}&\hline{\mathrm{253}}\\{\mathrm{23}}&\hline{\mathrm{276}<\mathrm{300}\:\left(\mathrm{6}\right)}\\{\mathrm{24}}&\hline{\mathrm{300}}\\{\mathrm{25}}&\hline{\mathrm{325}<\mathrm{350}\:\left(\mathrm{7}\right)}\\{\mathrm{26}}&\hline{\mathrm{351}}\\{\mathrm{27}}&\hline{\mathrm{378}<\mathrm{400}\:\left(\mathrm{8}\right)}\\{\mathrm{28}}&\hline{\mathrm{406}}\\{\mathrm{29}}&\hline{\mathrm{435}<\mathrm{450}\:\left(\mathrm{9}\right)}\\{\mathrm{30}}&\hline{\mathrm{465}}\\{\mathrm{31}}&\hline{\mathrm{496}<\mathrm{500}\:\left(\mathrm{10}\right)}\\{\mathrm{32}}&\hline{\mathrm{528}<\mathrm{550}\:\left(\mathrm{11}\right)}\\{\mathrm{33}}&\hline{\mathrm{561}}\\{\mathrm{34}}&\hline{\mathrm{595}<\mathrm{600}\:\left(\mathrm{12}\right)}\\{\mathrm{35}}&\hline{\mathrm{630}<\mathrm{650}\:\left(\mathrm{13}\right)}\\{\mathrm{36}}&\hline{\mathrm{666}<\mathrm{700}\:\left(\mathrm{14}\right)}\\{\mathrm{37}}&\hline{\mathrm{703}}\\{\mathrm{38}}&\hline{\mathrm{741}<\mathrm{750}\:\left(\mathrm{15}\right)}\\{\mathrm{39}}&\hline{\mathrm{780}<\mathrm{800}\:\left(\mathrm{16}\right.}\\{\mathrm{40}}&\hline{\mathrm{820}<\mathrm{850}\:\left(\mathrm{17}\right)}\\{\mathrm{41}}&\hline{\mathrm{861}<\mathrm{900}\:\left(\mathrm{18}\right)}\\{\mathrm{42}}&\hline{\mathrm{903}}\\{\mathrm{43}}&\hline{\mathrm{946}<\mathrm{950}\:\left(\mathrm{19}\right)}\\{\mathrm{44}}&\hline{\mathrm{990}<\mathrm{1000}\:\left(\mathrm{20}\right)}\\{\mathrm{45}}&\hline{\mathrm{1035}<\mathrm{1050}\:\left(\mathrm{21}\right)}\\{\mathrm{46}}&\hline{\mathrm{1081}<\mathrm{1100}\:\left(\mathrm{22}\right)}\\{\mathrm{47}}&\hline{\mathrm{1128}<\mathrm{1150}\:\left(\mathrm{23}\right)}\\{\mathrm{48}}&\hline{\mathrm{1176}<\mathrm{1200}\:\left(\mathrm{24}\right)}\\{\mathrm{49}}&\hline{\mathrm{1225}<\mathrm{1250}\:\left(\mathrm{25}\right)}\\{\mathrm{50}}&\hline{\mathrm{1275}<\mathrm{1300}\:\left(\mathrm{26}\right)}\\\hline\end{array}\:\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}{{n}}&\hline{\underset{\overset{\mathrm{OR}} {{cumulative}\:{total}}} {{n}\left({n}+\mathrm{1}\right)/\mathrm{2}}}\\{\mathrm{51}}&\hline{\mathrm{1326}<\mathrm{1350}\:\left(\mathrm{27}\right)}\\{\mathrm{52}}&\hline{\mathrm{1378}<\mathrm{1400}\:\left(\mathrm{28}\right)}\\{\mathrm{53}}&\hline{\mathrm{1431}<\mathrm{1450}\:\left(\mathrm{29}\right)}\\{\mathrm{54}}&\hline{\mathrm{1485}<\mathrm{1500}\:\left(\mathrm{30}\right.}\\{\mathrm{55}}&\hline{\mathrm{1540}<\mathrm{1550}\:\left(\mathrm{31}\right)}\\{\mathrm{56}}&\hline{\mathrm{1596}<\mathrm{1600}\:\left(\mathrm{32}\right)}\\{\mathrm{57}}&\hline{\mathrm{1653}<\mathrm{1700}\:\left(\mathrm{33}\right)}\\{\mathrm{58}}&\hline{\mathrm{1711}<\mathrm{1750}\:\left(\mathrm{34}\right)}\\{\mathrm{59}}&\hline{\mathrm{1770}<\mathrm{1800}\:\left(\mathrm{35}\right)}\\{\mathrm{60}}&\hline{\mathrm{1830}<\mathrm{1850}\:\left(\mathrm{36}\right)}\\{\mathrm{61}}&\hline{\mathrm{1891}<\mathrm{1900}\:\left(\mathrm{37}\right)}\\{\mathrm{62}}&\hline{\mathrm{1953}<\mathrm{2000}\:\left(\mathrm{38}\right)}\\{\mathrm{63}}&\hline{\mathrm{2016}<\mathrm{2050}\:\left(\mathrm{39}\right)}\\{\mathrm{64}}&\hline{\mathrm{2080}<\mathrm{2100}\:\left(\mathrm{40}\right)}\\{\mathrm{65}}&\hline{\mathrm{2145}<\mathrm{2150}\:\left(\mathrm{41}\right)}\\{\mathrm{66}}&\hline{\mathrm{2211}<\mathrm{2250}\:\left(\mathrm{42}\right)}\\{\mathrm{67}}&\hline{\mathrm{2278}<\mathrm{2300}\:\left(\mathrm{43}\right)}\\{\mathrm{68}}&\hline{\mathrm{2346}<\mathrm{2350}\:\left(\mathrm{44}\right)}\\{\mathrm{69}}&\hline{\mathrm{2415}<\mathrm{2450}\:\left(\mathrm{45}\right)}\\{\mathrm{70}}&\hline{\mathrm{2485}<\mathrm{2500}\:\left(\mathrm{46}\right)}\\{\mathrm{71}}&\hline{\mathrm{2556}<\mathrm{2600}\:\left(\mathrm{47}\right)}\\{\mathrm{72}}&\hline{\mathrm{2628}<\mathrm{2650}\:\left(\mathrm{48}\right)}\\{\mathrm{73}}&\hline{\mathrm{2701}<\mathrm{2750}\:\left(\mathrm{49}\right)}\\{\mathrm{74}}&\hline{\mathrm{2775}<\mathrm{2800}\:\left(\mathrm{50}\right)}\\{\mathrm{75}}&\hline{\mathrm{2850}<\mathrm{2900}\:\left(\mathrm{51}\right)}\\{\mathrm{76}}&\hline{\mathrm{2926}<\mathrm{2950}\:\left(\mathrm{52}\right)}\\{\mathrm{77}}&\hline{\mathrm{3003}<\mathrm{3050}\:\left(\mathrm{53}\right)}\\{\mathrm{78}}&\hline{\mathrm{3081}<\mathrm{3100}\:\left(\mathrm{54}\right)}\\{\mathrm{79}}&\hline{\mathrm{3160}<\mathrm{3200}\:\left(\mathrm{55}\right)}\\{\mathrm{80}}&\hline{\mathrm{3240}<\mathrm{3250}\:\left(\mathrm{56}\right)}\\{\mathrm{81}}&\hline{\mathrm{3321}<\mathrm{3350}\:\left(\mathrm{57}\right)}\\{\mathrm{82}}&\hline{\mathrm{3403}<\mathrm{3450}\:\left(\mathrm{58}\right)}\\{\mathrm{83}}&\hline{\mathrm{3486}<\mathrm{3500}\:\left(\mathrm{59}\right)}\\{\mathrm{84}}&\hline{\mathrm{3570}<\mathrm{3600}\:\left(\mathrm{60}\right)}\\{\mathrm{85}}&\hline{\mathrm{3655}<\mathrm{3700}\:\left(\mathrm{61}\right)}\\{\mathrm{86}}&\hline{\mathrm{3741}<\mathrm{3750}\:\left(\mathrm{62}\right)}\\{\mathrm{87}}&\hline{\mathrm{3828}<\mathrm{3850}\:\left(\mathrm{63}\right)}\\{\mathrm{88}}&\hline{\mathrm{3916}<\mathrm{3950}\:\left(\mathrm{64}\right)}\\{\mathrm{89}}&\hline{\mathrm{4005}<\mathrm{4050}\:\left(\mathrm{65}\right)}\\{\mathrm{90}}&\hline{\mathrm{4095}<\mathrm{4100}\:\left(\mathrm{66}\right)}\\{\mathrm{91}}&\hline{\mathrm{4186}<\mathrm{4200}\:\left(\mathrm{67}\right)}\\{\mathrm{92}}&\hline{\mathrm{4278}<\mathrm{4300}\:\left(\mathrm{68}\right)}\\{\mathrm{93}}&\hline{\mathrm{4371}<\mathrm{4400}\:\left(\mathrm{69}\right)}\\{\mathrm{94}}&\hline{\mathrm{4465}<\mathrm{4500}\:\left(\mathrm{70}\right)}\\{\mathrm{95}}&\hline{\mathrm{4560}<\mathrm{4600}\:\left(\mathrm{71}\right)}\\{\mathrm{96}}&\hline{\mathrm{4656}<\mathrm{4700}\:\left(\mathrm{72}\right)}\\{\mathrm{97}}&\hline{\mathrm{4753}<\mathrm{4800}\:\left(\mathrm{73}\right)}\\{\mathrm{98}}&\hline{\mathrm{4851}<\mathrm{4900}\:\left(\mathrm{74}\right)}\\{\mathrm{99}}&\hline{\mathrm{4950}<\mathrm{5000}\:\left(\mathrm{75}\right)}\\{\mathrm{100}}&\hline{\underset{\begin{array}{|c|}{\:\:\:\:\:\mathrm{76}\:\:\:\:\:}\\\hline\end{array}} {\mathrm{5050}<\mathrm{5100}\:\left(\mathrm{76}\right)}}\\\hline\end{array} \\ $$$${Changing}\:{colour}\:\:{of}\:{font}\:{here}\: \\ $$$${means}\:'{other}\:{distinct}\:{integer}'. \\ $$
Commented by mr W last updated on 15/Sep/22
sum(if(ceil(a(k))<a(k+1)),k=0..101)
$$\mathrm{sum}\left(\mathrm{if}\left(\mathrm{ceil}\left(\mathrm{a}\left(\mathrm{k}\right)\right)<\mathrm{a}\left(\mathrm{k}+\mathrm{1}\right)\right),\mathrm{k}=\mathrm{0}..\mathrm{101}\right) \\ $$
Commented by Rasheed.Sindhi last updated on 15/Sep/22
Ans: 76  adhigenz sir, please confirm the  answer before I give final shape to  my solution.
$${Ans}:\:\mathrm{76} \\ $$$${adhigenz}\:{sir},\:{please}\:{confirm}\:{the} \\ $$$${answer}\:{before}\:{I}\:{give}\:{final}\:{shape}\:{to} \\ $$$${my}\:{solution}. \\ $$
Commented by mr W last updated on 15/Sep/22
answer 76 is correct sir.
$${answer}\:\mathrm{76}\:{is}\:{correct}\:{sir}. \\ $$
Commented by Rasheed.Sindhi last updated on 15/Sep/22
T h a n k s   sir!
$$\mathbb{T}\:\boldsymbol{\mathrm{h}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{n}}\:\Bbbk\:\boldsymbol{\mathrm{s}}\:\:\:\boldsymbol{\mathrm{sir}}! \\ $$
Commented by mr W last updated on 15/Sep/22
⌊((n^2 +n)/(100))⌋=k  1≤n≤100  ⇒0≤k≤101  but not each number from 0 to 101 is  for k possible.    k≤((n(n+1))/(100))<k+1  100k≤n(n+1)<100(k+1)  n^2 +n−100k≥0  n≥((−1+(√(1+400k)))/2)  n^2 +n−100(k+1)<0  n<((−1+(√(1+400(k+1))))/2)  ((−1+(√(1+400k)))/2)≤n<((−1+(√(1+400(k+1))))/2)  we can see a “k” is possible, i.e. a “n”  exists, only if  ⌈((−1+(√(1+400k)))/2)⌉<((−1+(√(1+400(k+1))))/2)  for k from 0 to 101, only 76 numbers  fulfill this condition. i checked this  using Graph.
$$\lfloor\frac{{n}^{\mathrm{2}} +{n}}{\mathrm{100}}\rfloor={k} \\ $$$$\mathrm{1}\leqslant{n}\leqslant\mathrm{100} \\ $$$$\Rightarrow\mathrm{0}\leqslant{k}\leqslant\mathrm{101} \\ $$$${but}\:{not}\:{each}\:{number}\:{from}\:\mathrm{0}\:{to}\:\mathrm{101}\:{is} \\ $$$${for}\:{k}\:{possible}. \\ $$$$ \\ $$$${k}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{100}}<{k}+\mathrm{1} \\ $$$$\mathrm{100}{k}\leqslant{n}\left({n}+\mathrm{1}\right)<\mathrm{100}\left({k}+\mathrm{1}\right) \\ $$$${n}^{\mathrm{2}} +{n}−\mathrm{100}{k}\geqslant\mathrm{0} \\ $$$${n}\geqslant\frac{−\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{400}{k}}}{\mathrm{2}} \\ $$$${n}^{\mathrm{2}} +{n}−\mathrm{100}\left({k}+\mathrm{1}\right)<\mathrm{0} \\ $$$${n}<\frac{−\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{400}\left({k}+\mathrm{1}\right)}}{\mathrm{2}} \\ $$$$\frac{−\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{400}{k}}}{\mathrm{2}}\leqslant{n}<\frac{−\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{400}\left({k}+\mathrm{1}\right)}}{\mathrm{2}} \\ $$$${we}\:{can}\:{see}\:{a}\:“{k}''\:{is}\:{possible},\:{i}.{e}.\:{a}\:“{n}'' \\ $$$${exists},\:{only}\:{if} \\ $$$$\lceil\frac{−\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{400}{k}}}{\mathrm{2}}\rceil<\frac{−\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{400}\left({k}+\mathrm{1}\right)}}{\mathrm{2}} \\ $$$${for}\:{k}\:{from}\:\mathrm{0}\:{to}\:\mathrm{101},\:{only}\:\mathrm{76}\:{numbers} \\ $$$${fulfill}\:{this}\:{condition}.\:{i}\:{checked}\:{this} \\ $$$${using}\:{Graph}. \\ $$
Commented by mr W last updated on 15/Sep/22
Commented by mr W last updated on 15/Sep/22
you did a nice work sir using the table!
$${you}\:{did}\:{a}\:{nice}\:{work}\:{sir}\:{using}\:{the}\:{table}! \\ $$
Commented by Rasheed.Sindhi last updated on 15/Sep/22
T_ ^(H^A N) X                     S_I R
$$\mathbb{T}_{} ^{\mathbb{H}^{\mathbb{A}} \mathbb{N}} \mathbb{X}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\mathcal{S}_{\mathcal{I}} \mathcal{R} \\ $$
Commented by Tawa11 last updated on 18/Sep/22
Great sirs
$$\mathrm{Great}\:\mathrm{sirs} \\ $$

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