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S-1-1-1-2-2-3-3-16-16-S-2-1-1-2-2-3-3-14-14-Find-S-1-S-2-

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Question Number 223571 by hardmath last updated on 30/Jul/25
S_1 = 1∙1! + 2∙2! + 3∙3! +...+ 16∙16! S_2 = 1∙1! + 2∙2! + 3∙3! +...+ 14∙14! Find: (S_1 /S_2 ) = ?
$$\mathrm{S}_{\mathrm{1}} \:=\:\mathrm{1}\centerdot\mathrm{1}!\:+\:\mathrm{2}\centerdot\mathrm{2}!\:+\:\mathrm{3}\centerdot\mathrm{3}!\:+…+\:\mathrm{16}\centerdot\mathrm{16}! \\ $$$$\mathrm{S}_{\mathrm{2}} \:=\:\mathrm{1}\centerdot\mathrm{1}!\:+\:\mathrm{2}\centerdot\mathrm{2}!\:+\:\mathrm{3}\centerdot\mathrm{3}!\:+…+\:\mathrm{14}\centerdot\mathrm{14}! \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{S}_{\mathrm{1}} }{\mathrm{S}_{\mathrm{2}} }\:=\:? \\ $$
Answered by parthasc last updated on 30/Jul/25
Answered by Raphael254 last updated on 01/Sep/25
S_(1 ) = Σ_(i = 1) ^(16) i×i! S_2 = Σ_(i = 1) ^(14) i×i! See a curious thing: ((4!)/(1×1! + 2×2! + 3×3!)) = ((24)/(1 + 4 + 18)) = ((24)/(23)) ((5!)/(1×1! + 2×2! + 3×3! ×4×4!)) = ((5!)/(1 + 4 + 18 + 96)) = ((120)/(119)) ((6!)/(1×1! × 2×2! + 3×3! × 4×4! + 5×5!)) = ((6!)/(1 + 4 + 18 + 96 + 600)) = ((720)/(719)) ((n!)/((n−1)(n−1)! + (n−2)(n−2)! + (n−3)(n−3)! + ... + (n−n)(n−n)!)) = ((n!)/(n! − 1)), n ≥ 2 Proof: (n−1)(n−1)! + (n−2)(n−2)! + (n−3)(n−3)! + ... + (n−n)(n−n))! (to be easy to visualize, we can invert to a crescent order) = (n − n)(n − n)! + (n−(n−1))(n−(n−1))! + (n−(n−2))(n−(n−2))! + ... + (n−(n− (n−1)))(n−(n−(n−1)))! = = 0×0! + (n − n + 1)(n−n+1)! + (n−n+2)(n−n+2)! + ... + (n−(n−n+1))(n−(n−n+1))! = 0 + 1×1! + 2×2! + ... + (n−1)(n−1)! = (1! − 1) + 1×1! + 2×2! + ... + (n−1)(n−1)! = 1! + 1×1! + 2×2! + ... + (n−1)(n−1)! − 1 = 1!(1 + 1) + 2×2! + ... + (n−1)(n−1)! − 1 = 1!(2) + 2×2! + ... + (n−1)(n−1)! − 1 = 2! + 2×2! + ... + (n−1)(n−1)! − 1 = 2!(1 + 2) + ... + (n−1)(n−1)! − 1 = 2!(3) + ... + (n−1)(n−1)! − 1 = 3! + ... + (n−1)(n−1)! − 1 = ... (n−2)!(n−1) + (n−1)(n−1)! − 1 = ... (n−1)! + (n−1)(n−1)! − 1 = ... (n−1)!(1 + (n−1)) − 1 = ... (n−1)!(1 + n − 1) − 1 = ... (n−1)!(n) − 1 = ... n! − 1 (S_1 /S_2 ) = ((Σ_(i = 1) ^(16) i×i!)/(Σ_(i = 1) ^(14) i×i!)) = ((1×1! ×2×2! + 3×3! + ... + 16×16!)/(1×1! + 2×2! + 3×3! + ... + 14×14!)) (((n +m)/n) = (n/n) + (m/n) = 1 + (m/n)) = ((1×1! + 2×2! + 3×3! + ... 14×14!)/(1×1! + 2×2! + 3×3! + ... 14×14!)) + ((15×15! + 16×16!)/(1×1! + 2×2! + 3×3! + ... + 14×14!)) = 1 + ((15×15! + 16×16!)/(1×1! + 2×2! + 3×3! + ... 14×14!)) = 1 + ((15×15! + 16×16×15!)/(1×1! + 2×2! + 3×3! + ... + 14×14!)) = 1 + ((15!(15 + 16^2 ))/(15! − 1)) ((15! − 1 + 15!(271))/(15! − 1)) = ((15!(1 + 271) − 1)/(15! − 1)) = ((15!(272) − 1)/(15! − 1)) (((m×n − 1)/(m − 1)) = n + ((n − 1)/(m − 1)) = n + ((m×(n/m) − 1)/(m − 1)) = n + ((n/m) + (((n/m) − 1)/(m − 1))) = n + ((n/m) + ((m×(n/(m×m)) − 1)/(m − 1))) = n + (n/m) + ((n/m^2 ) + (((n/m^2 ) − 1)/(m − 1))) = ... = n + (n/m) + (n/m^2 ) + ... + (n/m^p ) + (the next and last term can be introduced at any time after the first term ′n′ is introduced) (((n/m^p ) −1)/(m − 1)), p ≥ 0 and m ≠ 1); observation: in case of (((n/m^0 ) − 1)/(m − 1)), consider only ((n − 1)/(m − 1)), because when m = 0, we will have (((n/0^0 ) − 1)/(0 − 1)) which is undetermined, but the formula is still working in this case for the two first terms (n and ((n − 1)/(m − 1)), since a third term wouldn′t work, because the second term would be (n/m)). ((15!(272) − 1)/(15! − 1)) = 272 + (after n = 272) ((272 − 1)/(15! − 1)) = 272 + ((271)/(15! − 1)) ≈272 Suppose we wanted to add one more term: ((15!(272) − 1)/(15! − 1)) = 272 + ((272)/(15!)) + ((((272)/(15!)) − 1)/(15! − 1)) = 272 + ((272)/(15!)) + (((272 − 15!)/(15!))/(15! − 1)) = 272 + ((272)/(15!)) + ((272 − 15!)/(15!(15! − 1))) = 272 + ((272(15! − 1) + 272 − 15!)/(15!(15! −1))) 272 + ((272×15! − 272 + 272 − 15!)/(15!×15! − 15!)) = 272 + ((272×15! − 15!)/(15!×15! − 15!)) = 272 + ((15!(272 −1))/(15!(15! − 1))) = 272 + ((271)/(15! − 1)) (((271)/(15! − 1)) ≈ 0) Conclusion: (S_1 /S_2 ) = 272 + ((271)/(15! − 1)) ≈272
$$ \\ $$$${S}_{\mathrm{1}\:} =\:\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{16}} {\sum}}\:{i}×{i}! \\ $$$${S}_{\mathrm{2}} \:=\:\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{14}} {\sum}}\:{i}×{i}! \\ $$$$ \\ $$$${See}\:{a}\:{curious}\:{thing}: \\ $$$$ \\ $$$$\frac{\mathrm{4}!}{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!}\:=\:\frac{\mathrm{24}}{\mathrm{1}\:+\:\mathrm{4}\:+\:\mathrm{18}}\:=\:\frac{\mathrm{24}}{\mathrm{23}} \\ $$$$ \\ $$$$\frac{\mathrm{5}!}{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:×\mathrm{4}×\mathrm{4}!}\:=\:\frac{\mathrm{5}!}{\mathrm{1}\:+\:\mathrm{4}\:+\:\mathrm{18}\:+\:\mathrm{96}}\:=\:\frac{\mathrm{120}}{\mathrm{119}} \\ $$$$ \\ $$$$\frac{\mathrm{6}!}{\mathrm{1}×\mathrm{1}!\:×\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:×\:\mathrm{4}×\mathrm{4}!\:+\:\mathrm{5}×\mathrm{5}!}\:=\:\frac{\mathrm{6}!}{\mathrm{1}\:+\:\mathrm{4}\:+\:\mathrm{18}\:+\:\mathrm{96}\:+\:\mathrm{600}}\:=\:\frac{\mathrm{720}}{\mathrm{719}} \\ $$$$ \\ $$$$\frac{{n}!}{\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:+\:\left({n}−\mathrm{2}\right)\left({n}−\mathrm{2}\right)!\:+\:\left({n}−\mathrm{3}\right)\left({n}−\mathrm{3}\right)!\:+\:…\:+\:\left({n}−{n}\right)\left({n}−{n}\right)!}\:=\:\frac{{n}!}{{n}!\:−\:\mathrm{1}},\:{n}\:\geq\:\mathrm{2} \\ $$$$ \\ $$$${Proof}: \\ $$$$ \\ $$$$\left.\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:+\:\left({n}−\mathrm{2}\right)\left({n}−\mathrm{2}\right)!\:+\:\left({n}−\mathrm{3}\right)\left({n}−\mathrm{3}\right)!\:+\:…\:+\:\left({n}−{n}\right)\left({n}−{n}\right)\right)!\:\left({to}\:{be}\:{easy}\:{to}\:{visualize},\:{we}\:{can}\:{invert}\:{to}\:{a}\:{crescent}\:{order}\right) \\ $$$$=\:\left({n}\:−\:{n}\right)\left({n}\:−\:{n}\right)!\:+\:\left({n}−\left({n}−\mathrm{1}\right)\right)\left({n}−\left({n}−\mathrm{1}\right)\right)!\:+\:\left({n}−\left({n}−\mathrm{2}\right)\right)\left({n}−\left({n}−\mathrm{2}\right)\right)!\:+\:…\:+\:\left({n}−\left({n}−\:\left({n}−\mathrm{1}\right)\right)\right)\left({n}−\left({n}−\left({n}−\mathrm{1}\right)\right)\right)!\:= \\ $$$$=\:\mathrm{0}×\mathrm{0}!\:+\:\left({n}\:−\:{n}\:+\:\mathrm{1}\right)\left({n}−{n}+\mathrm{1}\right)!\:+\:\left({n}−{n}+\mathrm{2}\right)\left({n}−{n}+\mathrm{2}\right)!\:+\:…\:+\:\left({n}−\left({n}−{n}+\mathrm{1}\right)\right)\left({n}−\left({n}−{n}+\mathrm{1}\right)\right)! \\ $$$$=\:\mathrm{0}\:+\:\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)! \\ $$$$=\:\left(\mathrm{1}!\:−\:\mathrm{1}\right)\:+\:\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)! \\ $$$$=\:\mathrm{1}!\:+\:\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:\mathrm{1}!\left(\mathrm{1}\:+\:\mathrm{1}\right)\:+\:\mathrm{2}×\mathrm{2}!\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:\mathrm{1}!\left(\mathrm{2}\right)\:+\:\mathrm{2}×\mathrm{2}!\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:\mathrm{2}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:\mathrm{2}!\left(\mathrm{1}\:+\:\mathrm{2}\right)\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:\mathrm{2}!\left(\mathrm{3}\right)\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:\mathrm{3}!\:+\:…\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:…\:\left({n}−\mathrm{2}\right)!\left({n}−\mathrm{1}\right)\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:…\:\left({n}−\mathrm{1}\right)!\:\:+\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{1}\right)!\:−\:\mathrm{1} \\ $$$$=\:…\:\left({n}−\mathrm{1}\right)!\left(\mathrm{1}\:+\:\left({n}−\mathrm{1}\right)\right)\:−\:\mathrm{1} \\ $$$$=\:…\:\left({n}−\mathrm{1}\right)!\left(\mathrm{1}\:+\:{n}\:−\:\mathrm{1}\right)\:−\:\mathrm{1} \\ $$$$=\:…\:\left({n}−\mathrm{1}\right)!\left({n}\right)\:−\:\mathrm{1} \\ $$$$=\:…\:{n}!\:−\:\mathrm{1} \\ $$$$ \\ $$$$\frac{{S}_{\mathrm{1}} }{{S}_{\mathrm{2}} }\:=\:\frac{\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{16}} {\sum}}\:{i}×{i}!}{\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{14}} {\sum}}\:{i}×{i}!}\:=\:\frac{\mathrm{1}×\mathrm{1}!\:×\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:+\:…\:+\:\mathrm{16}×\mathrm{16}!}{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:+\:…\:+\:\mathrm{14}×\mathrm{14}!} \\ $$$$\left(\frac{{n}\:+{m}}{{n}}\:=\:\frac{{n}}{{n}}\:+\:\frac{{m}}{{n}}\:=\:\mathrm{1}\:+\:\frac{{m}}{{n}}\right)\:=\:\frac{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:+\:…\:\mathrm{14}×\mathrm{14}!}{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:+\:…\:\mathrm{14}×\mathrm{14}!}\:+\:\frac{\mathrm{15}×\mathrm{15}!\:+\:\mathrm{16}×\mathrm{16}!}{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:+\:…\:+\:\mathrm{14}×\mathrm{14}!}\:=\:\mathrm{1}\:+\:\frac{\mathrm{15}×\mathrm{15}!\:+\:\mathrm{16}×\mathrm{16}!}{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:+\:…\:\mathrm{14}×\mathrm{14}!}\:=\:\mathrm{1}\:+\:\frac{\mathrm{15}×\mathrm{15}!\:+\:\mathrm{16}×\mathrm{16}×\mathrm{15}!}{\mathrm{1}×\mathrm{1}!\:+\:\mathrm{2}×\mathrm{2}!\:+\:\mathrm{3}×\mathrm{3}!\:+\:…\:+\:\mathrm{14}×\mathrm{14}!}\:=\:\mathrm{1}\:+\:\frac{\mathrm{15}!\left(\mathrm{15}\:+\:\mathrm{16}^{\mathrm{2}} \right)}{\mathrm{15}!\:−\:\mathrm{1}} \\ $$$$\frac{\mathrm{15}!\:−\:\mathrm{1}\:+\:\mathrm{15}!\left(\mathrm{271}\right)}{\mathrm{15}!\:−\:\mathrm{1}}\:=\:\frac{\mathrm{15}!\left(\mathrm{1}\:+\:\mathrm{271}\right)\:−\:\mathrm{1}}{\mathrm{15}!\:−\:\mathrm{1}}\:=\:\frac{\mathrm{15}!\left(\mathrm{272}\right)\:−\:\mathrm{1}}{\mathrm{15}!\:−\:\mathrm{1}} \\ $$$$\left(\frac{{m}×{n}\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}}\:=\:{n}\:+\:\frac{{n}\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}}\:=\:{n}\:+\:\frac{{m}×\frac{{n}}{{m}}\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}}\:=\:{n}\:+\:\left(\frac{{n}}{{m}}\:+\:\frac{\frac{{n}}{{m}}\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}}\right)\:=\:{n}\:+\:\left(\frac{{n}}{{m}}\:+\:\frac{{m}×\frac{{n}}{{m}×{m}}\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}}\right)\:=\:{n}\:+\:\frac{{n}}{{m}}\:+\:\left(\frac{{n}}{{m}^{\mathrm{2}} }\:+\:\frac{\frac{{n}}{{m}^{\mathrm{2}} }\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}}\right)\:=\:…\:=\:{n}\:+\:\frac{{n}}{{m}}\:+\:\frac{{n}}{{m}^{\mathrm{2}} }\:+\:…\:+\:\frac{{n}}{{m}^{{p}} }\:+\:\left({the}\:{next}\:{and}\:{last}\:{term}\:{can}\:{be}\:{introduced}\:{at}\:{any}\:{time}\:{after}\:{the}\:{first}\:{term}\:'\boldsymbol{{n}}'\:{is}\:{introduced}\right)\:\frac{\frac{{n}}{{m}^{{p}} }\:−\mathrm{1}}{{m}\:−\:\mathrm{1}},\:{p}\:\geq\:\mathrm{0}\:{and}\:{m}\:\neq\:\mathrm{1}\right);\:{observation}:\:{in}\:{case}\:{of}\:\frac{\frac{{n}}{{m}^{\mathrm{0}} }\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}},\:{consider}\:{only}\:\frac{{n}\:−\:\mathrm{1}}{{m}\:−\:\mathrm{1}},\:{because}\:{when}\:{m}\:=\:\mathrm{0},\:{we}\:{will}\:{have}\:\frac{\frac{{n}}{\mathrm{0}^{\mathrm{0}} }\:−\:\mathrm{1}}{\mathrm{0}\:−\:\mathrm{1}}\:{which}\:{is}\:{undetermined},\:{but}\:{the}\:{formula}\:{is}\:{still}\:{working}\:{in}\:{this}\:{case}\:{for}\:{the}\:{two}\:{first}\:{terms}\:\left(\boldsymbol{{n}}\:{and}\:\frac{\boldsymbol{{n}}\:−\:\mathrm{1}}{\boldsymbol{{m}}\:−\:\mathrm{1}},\:{since}\:{a}\:{third}\:{term}\:{wouldn}'{t}\:{work},\:{because}\:{the}\:{second}\:{term}\:{would}\:{be}\:\frac{{n}}{{m}}\right). \\ $$$$\frac{\mathrm{15}!\left(\mathrm{272}\right)\:−\:\mathrm{1}}{\mathrm{15}!\:−\:\mathrm{1}}\:=\:\mathrm{272}\:+\:\left({after}\:{n}\:=\:\mathrm{272}\right)\:\frac{\mathrm{272}\:−\:\mathrm{1}}{\mathrm{15}!\:−\:\mathrm{1}}\:=\:\mathrm{272}\:+\:\frac{\mathrm{271}}{\mathrm{15}!\:−\:\mathrm{1}}\:\approx\mathrm{272} \\ $$$$ \\ $$$${Suppose}\:{we}\:{wanted}\:{to}\:{add}\:{one}\:{more}\:{term}: \\ $$$$ \\ $$$$\frac{\mathrm{15}!\left(\mathrm{272}\right)\:−\:\mathrm{1}}{\mathrm{15}!\:−\:\mathrm{1}}\:=\:\mathrm{272}\:+\:\frac{\mathrm{272}}{\mathrm{15}!}\:+\:\frac{\frac{\mathrm{272}}{\mathrm{15}!}\:−\:\mathrm{1}}{\mathrm{15}!\:−\:\mathrm{1}}\:=\:\mathrm{272}\:+\:\frac{\mathrm{272}}{\mathrm{15}!}\:+\:\frac{\frac{\mathrm{272}\:−\:\mathrm{15}!}{\mathrm{15}!}}{\mathrm{15}!\:−\:\mathrm{1}}\:=\:\mathrm{272}\:+\:\frac{\mathrm{272}}{\mathrm{15}!}\:+\:\frac{\mathrm{272}\:−\:\mathrm{15}!}{\mathrm{15}!\left(\mathrm{15}!\:−\:\mathrm{1}\right)}\:=\:\mathrm{272}\:+\:\frac{\mathrm{272}\left(\mathrm{15}!\:−\:\mathrm{1}\right)\:+\:\mathrm{272}\:−\:\mathrm{15}!}{\mathrm{15}!\left(\mathrm{15}!\:−\mathrm{1}\right)} \\ $$$$\mathrm{272}\:+\:\frac{\mathrm{272}×\mathrm{15}!\:−\:\mathrm{272}\:+\:\mathrm{272}\:−\:\mathrm{15}!}{\mathrm{15}!×\mathrm{15}!\:−\:\mathrm{15}!}\:=\:\mathrm{272}\:+\:\frac{\mathrm{272}×\mathrm{15}!\:−\:\mathrm{15}!}{\mathrm{15}!×\mathrm{15}!\:−\:\mathrm{15}!}\:=\:\mathrm{272}\:+\:\frac{\mathrm{15}!\left(\mathrm{272}\:−\mathrm{1}\right)}{\mathrm{15}!\left(\mathrm{15}!\:−\:\mathrm{1}\right)}\:=\:\mathrm{272}\:+\:\frac{\mathrm{271}}{\mathrm{15}!\:−\:\mathrm{1}}\:\left(\frac{\mathrm{271}}{\mathrm{15}!\:−\:\mathrm{1}}\:\approx\:\mathrm{0}\right) \\ $$$$ \\ $$$${Conclusion}: \\ $$$$ \\ $$$$\frac{{S}_{\mathrm{1}} }{{S}_{\mathrm{2}} }\:=\:\mathrm{272}\:+\:\frac{\mathrm{271}}{\mathrm{15}!\:−\:\mathrm{1}}\:\approx\mathrm{272} \\ $$
Commented by hardmath last updated on 31/Jul/25
cool dear professors thankyou
$$\mathrm{cool}\:\mathrm{dear}\:\mathrm{professors}\:\mathrm{thankyou} \\ $$

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