Question Number 194526 by mathlove last updated on 09/Jul/23
$$\frac{{f}\left({x}+\mathrm{1}\right)}{{f}\left({x}\right)}={x}^{\mathrm{2}\:\:\:\:\:\:\:\:} \:\:\:\:{f}\left({x}\right)=? \\ $$$$\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{3}\right)}=? \\ $$
Answered by JDamian last updated on 09/Jul/23
$$\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{3}\right)}=\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{3}\right)}×\frac{{f}\left(\mathrm{5}\right)}{{f}\left(\mathrm{5}\right)}×\frac{{f}\left(\mathrm{4}\right)}{{f}\left(\mathrm{4}\right)}= \\ $$$$=\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{5}\right)}×\frac{{f}\left(\mathrm{5}\right)}{{f}\left(\mathrm{4}\right)}×\frac{{f}\left(\mathrm{4}\right)}{{f}\left(\mathrm{3}\right)}=\mathrm{5}^{\mathrm{2}} ×\mathrm{4}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{2}} =\mathrm{3600} \\ $$
Answered by MM42 last updated on 09/Jul/23
$${if}\:\:{x}\in\mathbb{N}\Rightarrow{f}\left({n}+\mathrm{1}\right)={n}^{\mathrm{2}} {f}\left({n}\right)={n}^{\mathrm{2}} ×\left({n}−\mathrm{1}\right)^{\mathrm{2}} {f}\left({n}−\mathrm{1}\right) \\ $$$$=…=\left({n}!\right)^{\mathrm{2}} ×{f}\left(\mathrm{1}\right) \\ $$