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Question Number 111536 by Aina Samuel Temidayo last updated on 04/Sep/20

$$\mathrm{Let}2,3,5,6,7,10,11,...\mathrm{be}\mathrm{increasing}$$$$\mathrm{sequence}\mathrm{of}\mathrm{positive}\mathrm{integers}\mathrm{that}\mathrm{are}$$$$\mathrm{neither}\mathrm{the}\mathrm{square}\mathrm{nor}\mathrm{cube}\mathrm{of}\mathrm{an}$$$$\mathrm{integer}.\mathrm{Find}\mathrm{the}2016\mathrm{th}\mathrm{term}\mathrm{of}\mathrm{this}$$$$\mathrm{sequence}.$$

Commented by Rasheed.Sindhi last updated on 04/Sep/20

$$2070$$

Commented by Aina Samuel Temidayo last updated on 04/Sep/20

$$\mathrm{Your}\mathrm{solution}\mathrm{please}?$$

Answered by Lordose last updated on 04/Sep/20

$$\mathbf{The}\mathbf{sequence}\mathbf{is}\mathbf{an}\mathbf{A}.\mathbf{P}$$$${\mathbf{U}}_{\mathrm{n}}=\mathbf{a}+(\mathbf{n}-1)\mathbf{d}$$$$\mathbf{a}=2,\mathbf{d}=1\mathbf{n}=2016$$$${\mathbf{U}}_{\mathbf{n}}=2+2015=2017$$

Commented by Aina Samuel Temidayo last updated on 04/Sep/20

$$\mathrm{How}\mathrm{is}\mathrm{it}\mathrm{an}\mathrm{A}.\mathrm{P}?\mathrm{The}\mathrm{common}$$$$\mathrm{difference}\mathrm{is}\mathrm{not}\mathrm{uniform}.$$

Answered by Rasheed.Sindhi last updated on 06/Sep/20

$$\mid \{2,3,5,6,7,10,11,...\left(2016members\right)\}\mid$$$$=\mid \{1,2,3,...,n\}\mid$$$$-\mid \{1,4,9,...,{(\lfloor \sqrt{n}\rfloor )}^2\}\mid$$$$-\mid \{1,8,27,...,{(\lfloor \sqrt[3]{n}\rfloor )}^3\}\mid$$$$+\mid \{1,64,729,..{(\lfloor \sqrt[6]{n}\rfloor )}^6\}\mid$$$$2016=n-\lfloor \sqrt{n}\rfloor -\lfloor \sqrt[3]{n}\rfloor +\lfloor \sqrt[6]{n}\rfloor$$$$n=2070satisfytheaboveequation$$$$Hence2016thterm=2070$$

Commented by Aina Samuel Temidayo last updated on 06/Sep/20

$$\mathrm{Thanks}.$$

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