Let-2-3-5-6-7-10-11-be-increasing-sequence-of-positive-integers-that-are-neither-the-square-nor-cube-of-an-integer-Find-the-2016th-term-of-this-sequence-

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

Let 2, 3, 5, 6, 7, 10, 11, \dots be increasing

sequence of positive integers that are

neither the square nor cube of an

integer . Find the 2016 th term of this

sequence .

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

2070

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

Your solution please ?

Answered by Lordose last updated on 04/Sep/20

\boldsymbol The \boldsymbol sequence \boldsymbol is \boldsymbol an \boldsymbol A . \boldsymbol P

\boldsymbol U_{n} = \boldsymbol a + (\boldsymbol n - 1) \boldsymbol d

\boldsymbol a = 2, \boldsymbol d = 1 \boldsymbol n = 2016

\boldsymbol U_{\boldsymbol n} = 2 + 2015 = 2017

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

How is it an A . P ? The common

difference is not uniform .

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

∣ {2, 3, 5, 6, 7, 10, 11, \dots (2016 m e m b e r s)} ∣

=∣ {1, 2, 3, \dots, n} ∣

- ∣ {1, 4, 9, \dots, {(⌊ \sqrt{n} ⌋)}^{2}} ∣

- ∣ {1, 8, 27, \dots, {(⌊ \sqrt[3]{n} ⌋)}^{3}} ∣

+ ∣ {1, 64, 729, . . {(⌊ \sqrt[6]{n} ⌋)}^{6}} ∣

2016 = n - ⌊ \sqrt{n} ⌋ - ⌊ \sqrt[3]{n} ⌋ + ⌊ \sqrt[6]{n} ⌋

n = 2070 s a t i s f y t h e a b o v e e q u a t i o n

H e n c e 2016 t h t e r m = 2070

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

Thanks .

Thanks .