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An arithmetic sequence is a type of sequence where each term after the first is generated by adding a constant, called the common difference, to the previous term. For instance, if the first term of an arithmetic sequence is 3 and the common difference is 2, the sequence would be 3, 5, 7, 9, and so on.

Here’s how it works:

• The first term is denoted as \(a_1\).
• Every subsequent term is found by adding the common difference \(d\) to the previous term.
• The n-th term can be found using the formula: \(a_n = a_1 + (n-1) \times d\).

Notice that arithmetic sequences grow linearly. This means the difference between consecutive terms remains the same throughout. Understanding this type of sequence is crucial for solving various mathematical problems involving patterns and number arrangements.

A geometric sequence, unlike an arithmetic one, is defined by a constant ratio between successive terms, known as the common ratio. The sequence from the original exercise, for example, is geometric rather than arithmetic because it involves multiplying each term by \(2\). This constant multiplier characterizes geometric sequences.

Key characteristics include:

• Having a first term \(a_1\).
• A common ratio \(r\) where each term is the result of multiplying the previous term by \(r\).
• The n-th term given by the formula: \(a_n = a_1 \times r^{(n-1)}\).

This type of sequence is significantly different from an arithmetic sequence as terms can grow or shrink exponentially. Recognizing geometric sequences is fundamental in various fields, including finance, computer science, and biology, where exponential growth or decay is observed.

Understanding sequence formulas is essential for working efficiently with both arithmetic and geometric sequences, as well as other types. Sequence formulas can be of two main types: explicit and recursive.

Explicit formulas provide a direct way to find any term in the sequence without needing the previous term(s). These formulas are ideal for calculating terms quickly:

• For arithmetic sequences: \(a_n = a_1 + (n-1) \times d\)
• For geometric sequences: \(a_n = a_1 \times r^{(n-1)}\)

On the other hand, recursive formulas require knowledge of one or more preceding terms. They are helpful when you need to understand how a sequence unfolds:

• General form: \(a_n = f(a_{n-1})\), with an initial condition \(a_1\)

For the sequence \(-2.5, -5, -10, -20, -40, \dots\) from the exercise, the recursive formula is \(a_n = 2 \times a_{n-1}\) with \(a_1 = -2.5\). Mastery of both recursive and explicit formulas is vital for anyone delving into mathematical sequences.