# #a_n = sin(pi n/6)+cos(pi n)# ?

See below.

This is a periodic sequence in which

Attached a sequence plot

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To find the general term ( a_n ) for the given sequence, we have:

[ a_n = \sin\left(\frac{\pi n}{6}\right) + \cos(\pi n) ]

The terms involve trigonometric functions, and the values of ( \sin ) and ( \cos ) are periodic with period ( 2\pi ).

For ( \sin\left(\frac{\pi n}{6}\right) ): The period is ( 2\pi ), but in this case, the period becomes ( 12\pi ) due to the coefficient ( \frac{\pi}{6} ). Thus, for integer values of ( n ), ( \sin\left(\frac{\pi n}{6}\right) ) will repeat every 12 terms.

For ( \cos(\pi n) ): The cosine function has a period of ( 2\pi ). Thus, for integer values of ( n ), ( \cos(\pi n) ) will repeat every 2 terms.

To find the general term ( a_n ), we can determine the values for a few terms and observe the pattern:

For ( n = 0 ): [ a_0 = \sin(0) + \cos(0) = 0 + 1 = 1 ]

For ( n = 1 ): [ a_1 = \sin\left(\frac{\pi}{6}\right) + \cos(\pi) = \frac{1}{2} - 1 = -\frac{1}{2} ]

For ( n = 2 ): [ a_2 = \sin\left(\frac{\pi}{3}\right) + \cos(2\pi) = \frac{\sqrt{3}}{2} + 1 = \frac{\sqrt{3} + 2}{2} ]

By observing the pattern, we can generalize the terms as follows:

For ( n ) even: [ a_n = \sin\left(\frac{\pi n}{6}\right) + \cos(\pi n) ]

For ( n ) odd: [ a_n = \sin\left(\frac{\pi n}{6}\right) + \cos(\pi n) ]

Thus, the general term for the sequence ( a_n ) is: [ a_n = \sin\left(\frac{\pi n}{6}\right) + \cos(\pi n) ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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