Question Number 56426 by Hassen_Timol last updated on 16/Mar/19
$${Let}\:{consider}\:{the}\:{following}\:{sequence}\:: \\ $$$$ \\ $$$$\mathrm{1}\:,\:\mathrm{3}\:,\:\sqrt{\mathrm{17}}\:,\:\mathrm{5}\:,\:\sqrt{\mathrm{33}}\:,\:\sqrt{\mathrm{41}}\:,\:...\: \\ $$$$ \\ $$$${What}\:{may}\:{be}\:{the}\:{explicit}\:{formula}\:{that} \\ $$$${can}\:{gives}\:{this}\:{sequence}\:{of}\:{number}\:? \\ $$$$ \\ $$$${Thank}\:{you} \\ $$
Answered by kaivan.ahmadi last updated on 16/Mar/19
$$\sqrt{\mathrm{1}},\sqrt{\mathrm{9}},\sqrt{\mathrm{17}},\sqrt{\mathrm{25}},\sqrt{\mathrm{33}},\sqrt{\mathrm{41}},... \\ $$$${b}=\mathrm{1},{d}=\mathrm{8} \\ $$$${b}_{{n}} ={a}+\left({n}−\mathrm{1}\right){d}=\mathrm{1}+\left({n}−\mathrm{1}\right)\mathrm{8}=\mathrm{8}{n}−\mathrm{7} \\ $$$${a}_{{n}} =\sqrt{\mathrm{8}{n}−\mathrm{7}} \\ $$$$ \\ $$
Commented by Hassen_Timol last updated on 16/Mar/19
$${Thank}\:{you},\:{I}\:{didn}'{t}\:{think}\:{about}\:{this}\:{method}... \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19
$$\sqrt{\mathrm{1}}\:,\sqrt{\mathrm{9}}\:,\sqrt{\mathrm{17}}\:,\sqrt{\mathrm{25}}\:,\sqrt{\mathrm{33}}\:'\sqrt{\mathrm{41}}\:.. \\ $$$${T}_{{n}} =\sqrt{\mathrm{1}+\left({n}−\mathrm{1}\right)\mathrm{8}}\:\rightarrow\sqrt{\mathrm{8}{n}−\mathrm{7}}\: \\ $$