Question Number 82995 by M±th+et£s last updated on 26/Feb/20
Commented by mr W last updated on 26/Feb/20
$${a}_{\mathrm{1000}} =\frac{\mathrm{1}}{\mathrm{499501}}\:? \\ $$
Answered by mind is power last updated on 26/Feb/20
$$\Rightarrow\frac{\mathrm{1}}{{a}_{{n}+\mathrm{1}} }={n}+\frac{\mathrm{1}}{{a}_{{n}} } \\ $$$${let}\:{b}_{{n}} =\frac{\mathrm{1}}{{a}_{{n}} } \\ $$$$\Rightarrow{b}_{{n}+\mathrm{1}} ={n}+{b}_{{n}} \\ $$$${b}_{\mathrm{0}} =\mathrm{1} \\ $$$$\Rightarrow\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\left({b}_{{k}+\mathrm{1}} −{b}_{{k}} \right)=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}{k} \\ $$$$\Rightarrow{b}_{{n}} −{b}_{\mathrm{0}} =\frac{{n}}{\mathrm{2}}\left({n}−\mathrm{1}\right) \\ $$$$\Rightarrow{b}_{{n}} =\mathrm{1}+\frac{{n}}{\mathrm{2}}\left({n}−\mathrm{1}\right) \\ $$$$\Rightarrow{a}_{{n}} =\frac{\mathrm{1}}{{b}_{{n}} }=\frac{\mathrm{2}}{{n}^{\mathrm{2}} −{n}+\mathrm{2}} \\ $$$${a}_{\mathrm{1000}} =\frac{\mathrm{2}}{\mathrm{999002}}=\frac{\mathrm{1}}{\mathrm{499501}} \\ $$$$ \\ $$
Commented by mr W last updated on 26/Feb/20
$${fine}\:{sir}!\:{i}\:{got}\:{the}\:{same}. \\ $$
Commented by M±th+et£s last updated on 26/Feb/20
$${thank}\:{you}\:{sir}.{this}\:{is}\:{the}\:{value} \\ $$
Commented by mind is power last updated on 27/Feb/20
$${withe}\:{pleasur} \\ $$