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Discrete mathematics is an area of mathematics that is concerned with discrete, as opposed to continuous, structures. This includes objects that can only take on distinct, separated values, such as integers. Sequences, which are ordered lists of numbers, play a key role in discrete mathematics and are used to represent a series of quantities or elements that follow a specific pattern.

In our exercise, the sequence is a mathematical expression that produces a new number for each integer value of \(n\). Discrete mathematics covers a variety of sequences, including arithmetic, geometric, and more complex forms like the one in our exercise. These sequences often model real-world scenarios, such as population growth, computer algorithms, or financial predictions.

Understanding how to manipulate and analyze sequences is a fundamental skill in this field. It's not just about calculating terms; it's about recognizing patterns, proving properties, and applying these sequences to solve practical problems.

Exponential sequences are a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the 'common ratio'. In contrast with arithmetic sequences, where the difference between consecutive terms is constant, in exponential sequences, the ratio between consecutive terms is constant.

Exponential sequences can be recognized by their general form \(a \times r^n\), where \(a\) is the initial term, \(r\) is the common ratio, and \(n\) is the term position in the sequence. The sequence in our exercise, \(3 \times 2^n - 4 \times 5^n\), contains two exponential terms with bases 2 and 5, respectively. These sequences are often encountered in areas like compound interest, population dynamics, and radioactive decay.

Understanding the behavior of exponential sequences is crucial because they can model explosive growth or decay, and the values in the sequence can quickly become very large or very small. This property is fundamental to distinguishing them from other sequence types.

Sequence formulas provide a method to find the \(n\)-th term of a sequence without having to enumerate all previous terms. The sequence formula for any given problem is like a recipe that outlines the steps to follow to arrive at the desired term.

In our exercise, the sequence formula given is \(r_{n} = 3 \times 2^n - 4 \times 5^n\). To obtain any term in this sequence, you simply plug in the value of \(n\) into the formula and perform the arithmetic operations as shown in the step-by-step solution.

Importance of Understanding Sequence Formulas

• It allows for quick calculation of any term in the sequence without listing all terms.
• It provides insight into the nature of the sequence, whether it’s arithmetic, geometric, or another form.
• It assists in proving properties related to sequences, such as convergence and divergence.
• It's essential for solving complex problems in discrete mathematics and related fields.

Substituting and simplifying the terms as shown helps students not only to find the numerical answer but to grasp the process that underpins sequence operations.