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In a recursive sequence, the base case is the initial term or starting point. It anchors the sequence by providing a known value from which all other terms are derived.
The base case forms the foundation of the sequence, as every subsequent term relies on its value. When given a problem that involves recursive sequences, like in our exercise, identifying the base case is crucial.
For example, in the sequence that begins with 100, this initial value—100—is the base case.
This means no matter how complex the sequence becomes, it always rests on this first term. Think of the base case as the zero-step in a journey where you know exactly where you are starting. In more general cases, a recursive sequence may begin with one or several base cases.
Key points about base case:

• It's the known starting value.
• Serves as a reference point for calculating future terms.
• Without a base case, a recursive sequence cannot be properly defined.

A recursive formula is like a rulebook for generating new terms based on previous ones. In the context of recursive sequences, it determines how each term is related to the preceding term.
For our sequence with a constant multiplier of \(-1.6\), the recursive formula is defined as:\[a_{n} = -1.6 \times a_{n-1}\]This formula tells us: "To find the current term \(a_n\), multiply the previous term \(a_{n-1}\) by \(-1.6\)."This step is foundational to solving recursive sequences because it provides the method to transition from one term to the next.
It's important since it determines not only the numerical values but also the behavior and the pattern of the sequence. Recursive formulas are powerful tools because:

• They offer flexibility, allowing you to generate terms without having to compute every predecessor.
• Highlight the relationship between different parts of the sequence.

Each calculation acts as a domino, setting the stage for the next computation.

The constant multiplier in a recursive sequence refers to the fixed number by which each term is multiplied to generate the next term.
It is a crucial element because it directly affects the sequence's growth, behavior, and direction.In our example, the constant multiplier is \(-1.6\).
This means each term in the sequence is calculated by multiplying the previous term by \(-1.6\). The multiplier being a negative number implies that the sign of the terms flips as you progress. Understanding the constant multiplier allows you to predict:

• Whether the sequence will alternate in sign (positive to negative and vice versa).
• The potential rate of increase or decrease in the absolute value of sequence terms.
• The overall trend, whether the series is convergent (decreasing in magnitude) or divergent.

Constant multipliers can shine a light on more than individual calculations; they reveal the narrative of the sequence's progression.