Homework Help

Show the series an=n^2-n is strictly monotonic if n>=1?

user profile pic

yapayapa | Honors

Posted July 2, 2013 at 4:54 PM via web

dislike 1 like

Show the series an=n^2-n is strictly monotonic if n>=1?

Tagged with math, monotonic, series

1 Answer | Add Yours

user profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 2, 2013 at 5:01 PM (Answer #1)

dislike 1 like

You need to test if the given sequence is strictly increasing or decreasing, hence, you need to evaluate the difference of two consecutive members of the sequence, such that:

`a_(n+1) - a_n = (n+1)^2 - (n+1) - n^2 + n`

Raising to square yields:

`a_(n+1) - a_n = n^2 + 2n + 1 - n - 1 - n^2 + n`

Reducing dupliuacte members yields:

`a_(n+1) - a_n = 2n`

Since `n>=1` yields that `2n >= 2 > 0` , hence` a_(n+1) - a_n > 0` .

Hence, testing if the given sequence strictly increases or decreases yields that `a_(n+1) - a_n > 0 => a_(n+1) > a_n` , thus the sequence strictly increases.

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes