Problem 72 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} $$S=\frac{n}{2}\left[2 a_{1}+(n-1) d\right] \text { for } a_{1} \quad \text { (Mathematics) }$$

Step-by-Step Solution

Answer

\( a_1 = \frac{2S - n(n-1)d}{2n} \)

Step 1: Distribute the fraction

Multiply both sides of the equation by 2 to eliminate the fraction. This gives us: \[ 2S = n[2a_1 + (n-1)d] \]

Step 2: Expand the equation

Distribute \( n \) to the terms inside the brackets on the right side of the equation. This results in: \[ 2S = 2na_1 + n(n-1)d \]

Step 3: Isolate terms involving \( a_1 \)

Subtract \( n(n-1)d \) from both sides to isolate terms involving \( a_1 \) on one side:\[ 2S - n(n-1)d = 2na_1 \]

Step 4: Solve for \( a_1 \)

Divide both sides by \( 2n \) to solve for \( a_1 \):\[ a_1 = \frac{2S - n(n-1)d}{2n} \]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra, where the goal is to find the value of an unknown variable that makes the equation true. When you solve an equation, you perform a series of operations to isolate the variable on one side while keeping the equation balanced.
Understanding the properties of equality, such as adding, subtracting, multiplying, or dividing by the same number on both sides, is crucial in solving equations. For instance, in the given exercise, we began by eliminating fractions, a common first step, by multiplying both sides. This cleared the path to working with more manageable numbers.
Always ensure each step maintains the equation's balance, and remember that the same operations must be performed on both sides. Patience and precision in manipulating the equation guarantee you find the correct solution.

Formula Manipulation

Formula manipulation involves rearranging and rewriting formulas to solve for a specific variable. This skill is extremely useful in mathematics and science.
In the provided solution, we aimed to express the formula in terms of \( a_1 \), which required several algebraic steps. First, we distributed terms inside the brackets and used subtraction to isolate required terms. It’s crucial to understand methods such as distribution, factoring, and combining like terms for effective formula manipulation.
Let's not forget dividing both sides to finally solve for the desired variable. Practicing these algebraic manipulations can increase your understanding of how mathematical relationships are structured and interrelated.

Sequences and Series

Sequences and series are mathematical concepts that deal with ordered lists of numbers, often describing a pattern. In this context, the formula is related to arithmetic series, where terms are summed up to form a series.
An arithmetic sequence is characterized by having a constant difference between consecutive terms. This difference plays a vital role in forming the series. Understanding the structure of the formula helps in solving related problems effectively.

• Arithmetic sequences have the form: \(a_1, a_1 + d, a_1 + 2d, \ldots\)
• Arithmetic series is the sum of terms of a sequence.

This exercise teaches how formula manipulation can unveil relations within sequences and series, making it possible to find unknown variables like \( a_1 \) when other quantities are known.