# `a_n = sin(npi/6)` Determine whether the sequence with the given n'th term is monotonic and whether it is bounded.

Sine function is periodic function with period of `2pi.`

This means that the given sequence will have 12 unique values (because `12cdot pi/6=2pi`) and these values will repeat cyclically, more precisely `a_n=a_(n+12),` `forall n in NN.` Therefore, we conclude that the given sequence is not monotonic

On the other hand, codomain...

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Sine function is periodic function with period of `2pi.`

This means that the given sequence will have 12 unique values (because `12cdot pi/6=2pi`) and these values will repeat cyclically, more precisely `a_n=a_(n+12),` `forall n in NN.` Therefore, we conclude that the given sequence is not monotonic

On the other hand, codomain of the sine function is `[-1,1]` so the sequence is obviously bounded.

Maximum terms of the sequence are `a_(3+12k)=1,` `k in ZZ,` while the minimum terms are `a_(9+12k)=-1,` `k in ZZ.`

The image below shows the first 60 terms of the sequence. We can clearly see the periodic nature of the sequence.

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)