91Ó°ÊÓ

An arithmetic progression, also known as an arithmetic sequence, is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the "common difference." For example, in the sequence 2, 5, 8, 11, the common difference is 3 because each number increases by 3 from the previous one.
In the context of the problem, the function given is linear, expressed as \(g(x) = 12 - 4x\). This means each subsequent term in the series will decrease by 4, demonstrating a negative arithmetic progression with a common difference of -4.
The initial term is \(g(1) = 8\). Understanding that \(g(x)\) represents an arithmetic progression helps us apply specific methods to find sums efficiently, such as the series sum formula.

The series sum formula gives us a method to find the sum of terms in an arithmetic sequence. This formula is expressed as: \[S_n = \frac{n (a_1 + a_n)}{2}\]where:

• \(S_n\) is the sum of the series
• \(n\) is the number of terms
• \(a_1\) is the first term
• \(a_n\) is the last term

For the exercise, we have 50 terms, starting with \(g(1) = 8\) and ending with \(g(50) = -188\). Plugging these into the formula, you calculate the sum efficiently without needing to individually add each term. This method is particularly useful in finite mathematics and makes calculations much simpler and quicker.

Finite mathematics involves studying mathematical concepts applicable to finite or discrete sets of numbers. It covers topics like sequences, series, probability, and other concepts used in business, computer science, and social sciences.
In our problem, we're dealing with an arithmetic sequence, which is a finite set of numbers wherever only a defined number of terms are involved. This distinction is essential because it limits the number of calculations needed. For instance, without the finite limit of 50 terms, finding the sum of an infinite arithmetic progression would be more complex and require different mathematical tools.

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. In the given exercise, \(g(x) = 12 - 4x\) is an example of an algebraic expression. This expression represents a rule for generating the terms in the sequence.
The variable \(x\) serves as a placeholder that represents different numbers, allowing the function to describe a potentially infinite set of numbers compactly.
By substituting various values into this expression, you can generate different sequence elements, which then form the arithmetic progression explored in this exercise. This ability to compactly represent sequences and mathematical processes makes algebraic expressions a fundamental tool in both solving specific problems and exploring broader mathematical theories.