Question

State whether the sequence, whose nth terms are indicated, is bounded and whether it is eventually monotone, increasing, or decreasing. (Select all that apply.)

a. Bounded
b. Eventually Monotone Increasing
c. Eventually Monotone Decreasing
d. Not Applicable

Answer 1

The sequence is always positive and steadily decreasing, bounded, and eventually monotone decreasing. Hence, all the options are correct.

Step-by-step explanation:

The sequence given can be described as always positive, steadily decreasing. This means that each term in the sequence is positive and the values are steadily getting smaller as n increases. The sequence can be considered bounded because it does not grow infinitely large or become infinitely small.

Since the terms are always positive, the sequence has a lower bound of 0. However, it does not have an upper bound because it continues to decrease without limit. The sequence is also eventually monotone decreasing.

Eventually monotone means that after a certain point, the terms in the sequence consistently follow a pattern. In this case, the pattern is that the terms are always positive and steadily decreasing.