Question Number 56594 by Joel578 last updated on 19/Mar/19
$$\mathrm{Find}\:\mathrm{the}\:{n}\mathrm{th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence} \\ $$$$\left({a}\right)\:\frac{\mathrm{1}}{\mathrm{2}},\:\frac{\mathrm{1}}{\mathrm{4}},\:\frac{\mathrm{1}}{\mathrm{8}},\:\frac{\mathrm{7}}{\mathrm{62}},\:... \\ $$$$\left({b}\right)\:\frac{\mathrm{1}}{\mathrm{2}},\:\frac{\mathrm{1}}{\mathrm{4}},\:\frac{\mathrm{1}}{\mathrm{8}},\:\mathrm{0},\:... \\ $$
Answered by MJS last updated on 19/Mar/19
$$\mathrm{there}\:\mathrm{are}\:\mathrm{always}\:\infty\:\mathrm{possibilities} \\ $$$$\mathrm{with}\:\mathrm{four}\:\mathrm{terms}\:\mathrm{given},\:\mathrm{we}\:\mathrm{can}\:\mathrm{match}\:\mathrm{with} \\ $$$${x}_{{n}} ={c}_{\mathrm{3}} {n}^{\mathrm{3}} +{c}_{\mathrm{2}} {n}^{\mathrm{2}} +{c}_{\mathrm{1}} {n}+{c}_{\mathrm{0}} \\ $$$$\left(\mathrm{or}\:\mathrm{any}\:\mathrm{other}\:\mathrm{model}\:\mathrm{with}\:\mathrm{4}\:\mathrm{independent}\right. \\ $$$$\left.\mathrm{constants}\right) \\ $$$$\mathrm{we}\:\mathrm{get} \\ $$$$\left({a}\right)\:{x}_{{n}} =−\frac{\mathrm{1}}{\mathrm{496}}{n}^{\mathrm{3}} +\frac{\mathrm{37}}{\mathrm{496}}{n}^{\mathrm{2}} −\frac{\mathrm{57}}{\mathrm{124}}{n}+\frac{\mathrm{55}}{\mathrm{62}} \\ $$$$\left({b}\right)\:{x}_{{n}} =−\frac{\mathrm{1}}{\mathrm{48}}{n}^{\mathrm{3}} +\frac{\mathrm{3}}{\mathrm{16}}{n}^{\mathrm{2}} −\frac{\mathrm{2}}{\mathrm{3}}{n}+\mathrm{1} \\ $$
Commented by Joel578 last updated on 19/Mar/19
$${thank}\:{you}\:{sir}\:{MJS} \\ $$$${but},\:{which}\:{formula}\:\:{did}\:{u}\:{have}\:{used}? \\ $$$${I}\:{mean},\:{the}\:{chapter}\:{that}\:{studying}\:{this}\:{formula} \\ $$