Final Answer:
d) It alternates between increasing and decreasing because the sequence alternates between increasing and decreasing due to changes in the operations or values within the recursive formula. Thus the correct option is d) It alternates between increasing and decreasing.
Step-by-step explanation:
The sequence generated by a recursive formula alternates between increasing and decreasing when the successive terms display a pattern of fluctuations rather than consistently rising or falling. This can occur due to the alternation in the values or operations of subsequent terms in the sequence. For instance, the recursive formula might contain alternating additions and subtractions or multiplication and division, causing the sequence to fluctuate between increasing and decreasing patterns.
In such cases, upon evaluating the terms of the sequence using the recursive formula, one can observe a pattern where the sequence might rise initially, then decrease, followed by another increase, and so forth. This alternating behavior indicates that the sequence neither strictly ascends nor descends but rather fluctuates between rising and falling with each subsequent term.
Analyzing the recursive formula and observing the values of the generated sequence for a few initial terms can confirm this alternating pattern. A clear demonstration of fluctuating trends in the sequence's terms showcases the alternation between increasing and decreasing, providing evidence that supports the choice of the answer as the sequence exhibiting an alternating pattern rather than solely increasing, decreasing, or staying constant.
Thus the correct option is d) It alternates between increasing and decreasing.