is it or is it not monotonic?

Question

an=n/2n+2, Is this monotonic? Justify

(n in an is subscript)

Answer 1

I assume that you mean n/(2n+2). Well, a_(n+1)=(n+1)/(2(n+1) +2)
=(n+1)/(2n+4). So you want to compare n/(2n+2) with (n+1)/(2n+4). By cross multiplication, this compares n(2n+4) with (n+1)(2n+2), in other words 2n^2+4n with 2n^2 +4n +2.. Clearly the latter is larger, so a_n

Answer 2

For a_n to be monotonic, a_(n+1) - a_n can't change signals with varying n.

a_(n+1) - a_n =
(n+1)/(2(n+1)+2) - n/(2n+2) =
(n+1)/(2n+4) - n/(2n+2) =
[(n+1)(2n+2) - n(2n+4)] / (2n+4)(2n+2) =
(2n^2 + 4n + 2 - 2n^2 - 4n) / 4(n+2)(n+1) =
2 / 4(n+2)(n+1) =
1 / 2(n+2)(n+1)

Since n is always >= 0, 1 / 2(n+2)(n+1) is always positive. Thus, a_n is monotonic, and moreover, strictly increasing.