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U-n-2-3-5-6-7-8-10-11-12-13-14-find-U-n-




Question Number 7562 by Master Moon last updated on 03/Sep/16
(U_n )= 2,3,5,6,7,8,10,11,12,13,14,....  find U_n
$$\left({U}_{{n}} \right)=\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{10},\mathrm{11},\mathrm{12},\mathrm{13},\mathrm{14},…. \\ $$$${find}\:{U}_{{n}} \\ $$
Commented by Rasheed Soomro last updated on 03/Sep/16
Perfect square numbers have  been excluded from  natural numbers.
$${Perfect}\:{square}\:{numbers}\:{have}\:\:{been}\:{excluded}\:{from} \\ $$$${natural}\:{numbers}. \\ $$
Commented by 123456 last updated on 04/Sep/16
u_(n+1) = { ((u_n +1       (√(u_n +1))∉N)),((u_n +2        (√(u_n +1))∈N)) :}  u_1 =2  (n+1)^2 −n^2 =2n+1
$${u}_{{n}+\mathrm{1}} =\begin{cases}{{u}_{{n}} +\mathrm{1}\:\:\:\:\:\:\:\sqrt{{u}_{{n}} +\mathrm{1}}\notin\mathbb{N}}\\{{u}_{{n}} +\mathrm{2}\:\:\:\:\:\:\:\:\sqrt{{u}_{{n}} +\mathrm{1}}\in\mathbb{N}}\end{cases} \\ $$$${u}_{\mathrm{1}} =\mathrm{2} \\ $$$$\left({n}+\mathrm{1}\right)^{\mathrm{2}} −{n}^{\mathrm{2}} =\mathrm{2}{n}+\mathrm{1} \\ $$

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