Student Question

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

Expert Answers

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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.                                                                  

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