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Show that x_n=n+(n)

Source: ISI (BS) 2009 #10

May 5, 2012
number theory proposednumber theory

Problem Statement

Let xnx_n be the nn-th non-square positive integer. Thus x1=2,x2=3,x3=5,x4=6,x_1=2, x_2=3, x_3=5, x_4=6, etc. For a positive real number xx, denotes the integer closest to it by ⟨x⟩\langle x\rangle. If x=m+0.5x=m+0.5, where mm is an integer, then define ⟨x⟩=m\langle x\rangle=m. For example, ⟨1.2⟩=1,⟨2.8⟩=3,⟨3.5⟩=3\langle 1.2\rangle =1, \langle 2.8 \rangle =3, \langle 3.5\rangle =3. Show that xn=n+⟨n⟩x_n=n+\langle \sqrt{n}\rangle
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