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In algebra, a sequence is an ordered list of numbers that follow a certain rule. Each number in a sequence is called a term. In the context of the exercise, the sequence is defined by an algebraic expression, and the terms are generated based on their position in the sequence, which is denoted by the variable 'n'. For instance, the first term is denoted by 12-10(1), the second by 12-10(2), and so on.

When we describe a sequence, we often look for a pattern that can be applied to find any term in the sequence. In the provided exercise, noticing that each subsequent term decreases by 10 helps us understand the structure of the sequence. This observation leads us to formulate a rule that can generate the terms of the sequence, which can be either explicit (a direct formula for any term) or recursive (a step-by-step approach to find the next term).

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables, such as 'n' or 'x', and operators such as addition, subtraction, multiplication, and division. The expression describes a calculation that needs to be performed to find a value. For example, in the recursive rule algebra exercise, the expression given is \( a_n=12-10n \). This expression represents a rule for finding the 'n-th' term of a sequence by substituting the variable 'n' with the position of the term.

The beauty of algebraic expressions is that they offer a concise way to represent mathematical relationships and are the building blocks for forming algebraic equations. Understanding these expressions enables students to translate word problems into solvable equations and find patterns within sequences.

An arithmetic sequence is a type of sequence in which each term after the first is found by adding a constant value to the previous term. This constant value is known as the common difference, and it can be positive or negative. If you take any two successive terms from the sequence, their difference will always be the common difference.

In the case of our given exercise, the sequence \(a_n=12-10n\) is indeed an arithmetic sequence with a common difference of -10. This means that each term is 10 less than the term before it. Thus, to write a recursive rule for an arithmetic sequence, you consider the initial term and the common difference. Here, the recursive rule becomes \(a_n = a_{n-1} - 10\), signifying that to find any term in the sequence (except the first term), you subtract 10 from the preceding term. Understanding arithmetic sequences is crucial since they often model real-life situations where there is a consistent rate of change, such as the depreciation of a car's value over time or the balance of a savings account as regular deposits are made.