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In mathematics, a sequence represents a list of numbers arranged in a specific order. Each number in the sequence is referred to as a term. Understanding the concept of a recursive sequence can initially seem complex, but it's quite straightforward with practice. A recursive sequence is one where each term is dependent on the one before it. For example, the sequence described in the original exercise is recursive. It starts with an initial term, 100, and each subsequent term is found by multiplying the previous term by a fixed number, known in this exercise as the multiplier (-1.6).
To calculate the terms in this sequence, we use a simple process:

• Identify the starting term.
• Apply the recursive formula to find the next term, which is the previous term multiplied by the constant.
• Continue this process until you've calculated all desired terms.

This methodical approach is fundamental in solving recursive sequence problems, emphasizing the importance of the initial term and the recursive formula.

A multiplicative pattern is key to understanding how a recursive sequence develops. When given a constant multiplier like -1.6, the sequence will often alternate between positive and negative values, depending on the sign of the multiplier and the starting term. Here, each term changes the sequence's direction because multiplying a positive number by a negative results in a negative number, and vice versa.
This multiplication isn't random, and it follows a structured rule:

• Start with the initial term provided.
• Multiply each subsequent term by the multiplier to get the next term.
• Recognize patterns, such as alternating signs or growing magnitudes.

In this particular exercise, after the initial term of 100, the results are alternately negative and positive due to the (-1.6) multiplier, displaying a clear pattern that emerges when applying consistent multiplication.

Applying algebraic concepts is crucial when understanding recursive sequences. The recursive formula effectively utilizes algebra to provide structure and predictability in sequence calculations. The formula used in the exercise is:
\[ a_{n} = a_{n-1} \times (-1.6) \] In this formula, \( a_{n} \) represents the current term, and \( a_{n-1} \) is the previous term. The multiplier, \(-1.6\), dictates how each term relates to the last. Algebra helps in systematically approaching sequence calculations, ensuring that each step follows logically from the preceding one. This process illustrates valuable algebraic applications:

• Using expressions to represent patterns in sequences.
• Establishing relationships between numbers through operations.
• Making predictions about future terms based on the formula and initial conditions.

By breaking down problems using algebraic principles, solving for the terms in sequences becomes a manageable task that reinforces the utility of algebra in various mathematical scenarios.