Question

Define a bitonic sequence as one that first strictly increases and then immediately strictly decreases. Provide examples to illustrate this concept. For instance, [100, 200, 100), [100, 200, 300, 50), and [100, 200, 50, -100) are considered bitonic, as they exhibit the required pattern. In contrast, sequences like [1, 2, 2, 11], (1.2, 3), and [3, 2, 1) do not meet the bitonic criteria. Emphasize that a bitonic sequence must consist of a minimum of three values

Answer 1

A bitonic sequence strictly increases and then strictly decreases. Examples include sequences like [10, 20, 30, 25, 5] or [3, 7, 15, 8, 2], which demonstrate the bitonic pattern of rising to a peak and then falling. Such sequences are a useful concept for analyzing patterns where an optimal point is reached before a decline.

Step-by-step explanation:

A bitonic sequence is defined as a sequence of numbers that first strictly increases and then immediately strictly decreases. To illustrate this concept with an example, consider the sequence [10, 20, 30, 25, 5]. Here, the sequence starts at 10, goes up to 30, and then decreases to 5. This is a classic bitonic sequence as the sequence increases up to a certain point (the peak) and then decreases after that point.

Another example would be the sequence [3, 7, 15, 8, 2]. Initially, the sequence rises from 3 to 15, and then it falls from 15 to 2. In both examples provided, the key characteristic is the existence of a single peak that switches the pattern from increasing to decreasing.

It is essential to note that a bitonic sequence must contain at least three values to exhibit the increasing and then decreasing pattern. Sequences that do not strictly follow the increase-then-decrease pattern, such as [1, 2, 2, 11], are not considered bitonic because they contain repeated values (2, 2) which interrupt the strictly increasing nature before the decrease begins. Similarly, [1, 2, 3] and [3, 2, 1] cannot be bitonic as they consist of less than three values or do not show a strict increase before decreasing.

To solve the mathematical problem completely, let us provide an instance of a bitonic sequence following the pattern described when substituting products over time. Let's say the original product is a smartphone model 'X'. The first replacement is an upgraded model 'Y' with better specifications, and the second replacement is the even more advanced model 'Z'. This sequence [X, Y, Z, Y, X] demonstrates an initial increase in quality up to model 'Z' and then a decrease back to model 'X'.