Question

Give example of a monotone subsequence of the sequence Sn=cos(n.π/3)

Answer 1

A monotone subsequence is a subsequence that is either increasing or decreasing, which we define as follows:

• A sequence (S) is increasing if
  `S_n \geq S_m` whenever n>m.
• A sequence (S) is decreasing if
  `S_n \leq S_m` whenever n>m.

Here, we can create a subsequence by taking every sixth term:

• When n = 6,
`S_n = \cos(6\pi /3) = \cos(2\pi) = 1`

• When n = 12,
`S_n = \cos(12 \pi /3) = \cos(4\pi) = 1`,

and so on, and every such element will have a value of 1. This is a monotone subsequence.