In the range of [ 0

rynosluv101wopds

rynosluv101wopds

Answered question

2022-04-10

In the range of [ 0 , 1 ] R , construct a series which is strictly increasing and rational number and irrational numbers are alternatively appearing.

Answer & Explanation

Madalynn Acosta

Madalynn Acosta

Beginner2022-04-11Added 15 answers

You may use numbers of the form n n + 1 such as 2 3 for rational numbers and their geometric mean n n + 1 n + 1 n + 2 = n n + 2 in between.
This way, your sequence would start
1 2 < 1 3 < 2 3 < 1 2 < 3 4 < 3 5 < 4 5 < 2 3 < . . .
Starting at n = 0, with rational a ( 2 n ) and irrational a ( 2 n + 1 ), the closed form is
a ( n ) = 1 + ( 1 ) n 2 n + 2 n + 4 + 1 ( 1 ) n 2 n + 1 n + 5
Equivalently,
a ( n ) = ( 2 n + 3 + ( 1 ) n ) ( 2 n + 5 ( 1 ) n ) ( 2 n + 7 + ( 1 ) n ) ( 2 n + 9 ( 1 ) n ) = 2 n 2 + 8 n + 7 + ( 1 ) n 2 n 2 + 16 n + 31 + ( 1 ) n

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