Part 2/5:
Conversely, a sequence is termed decreasing if each term is greater than the subsequent term:
[ a_n > a_{n+1} \quad \text{for all } n \geq 1 ]
In this case, an example would be ( a_1 > a_2 > a_3 > a_4 > \ldots ).
A sequence is monotonic if it is either increasing or decreasing. The term "monotonic" comes from the prefix "mono," indicating that the sequence moves in a single direction—either exclusively upward or downward.
Examples of Monotonic Sequences
Example 1: A Decreasing Sequence
Consider the sequence defined by:
[ a_n = \frac{3}{n + 5} ]
To determine whether this sequence is increasing or decreasing, we observe that as ( n ) increases, the denominator ( n + 5 ) grows larger, resulting in smaller values for ( a_n ). Therefore, we can conclude that: