Question Number 159222 by physicstutes last updated on 14/Nov/21
$$\mathrm{find}\:\mathrm{an}\:\mathrm{explicit}\:\mathrm{formula}\:\mathrm{for} \\ $$$$\mathrm{the}\:\mathrm{sequence} \\ $$$$\frac{\mathrm{2}}{\mathrm{3}},\:\frac{\mathrm{4}}{\mathrm{5}},\:\frac{\mathrm{8}}{\mathrm{9}},\:\frac{\mathrm{16}}{\mathrm{17}},\:\frac{\mathrm{32}}{\mathrm{33}},\:… \\ $$
Commented by MJS_new last updated on 14/Nov/21
$${a}_{{n}} =\mathrm{1}−\frac{\mathrm{12}}{{n}^{\mathrm{4}} −\mathrm{6}{n}^{\mathrm{3}} +\mathrm{23}{n}^{\mathrm{2}} −\mathrm{18}{n}+\mathrm{36}} \\ $$$${a}_{{n}} =\frac{\mathrm{7}}{\mathrm{33660}}{n}^{\mathrm{4}} −\frac{\mathrm{13}}{\mathrm{16830}}{n}^{\mathrm{3}} −\frac{\mathrm{767}}{\mathrm{33660}}{n}^{\mathrm{2}} +\frac{\mathrm{3433}}{\mathrm{16830}}{n}+\frac{\mathrm{818}}{\mathrm{1683}} \\ $$$$… \\ $$
Answered by TheSupreme last updated on 14/Nov/21
$${a}_{{n}} =\frac{\mathrm{2}^{{n}} }{\mathrm{2}^{{n}} +\mathrm{1}}\:{is}\:{one}\:{of}\:{the}\:{infinite}\:{solutions} \\ $$$$ \\ $$