Problem 72 - A Graphical Approach to College Algebra

Question

$$\text { Solve each formula for the indicated variable.}$$ $$A=\frac{P i}{1-(1+i)^{-n}}, \text { for } n$$

Step-by-Step Solution

Answer

The value of \(n\) is \(n = \frac{\log \left(\frac{A-Pi}{A}\right)}{-\log(1+i)}\)."

Step 1: Isolate the denominator

Multiply both sides of the equation by the denominator \(1 - (1+i)^{-n}\) to eliminate the fraction. This gives us: \(A \cdot (1 - (1+i)^{-n}) = Pi\).

Step 2: Distribute \(A\)

Distribute \(A\) on the left side of the equation: \(A - A(1+i)^{-n} = Pi\).

Step 3: Isolate the term with \(n\)

Subtract \(A\) from both sides to get the expression containing \(n\) alone on one side: \(- A(1+i)^{-n} = Pi - A\).

Step 4: Divide by \(-A\)

Divide both sides by \(-A\) to solve for \((1+i)^{-n}\): \((1+i)^{-n} = \frac{A-Pi}{A}\).

Step 5: Solve for \(n\) using properties of exponents

Apply logarithms to both sides to solve for \(n\). Since \((1+i)^{-n} = \frac{A-Pi}{A}\), applying the logarithm gives \(-n \log(1+i) = \log \left(\frac{A-Pi}{A}\right)\).

Step 6: Final expression for \(n\)

Solve for \(n\) by dividing both sides by \(-\log(1+i)\): \(n = \frac{\log \left(\frac{A-Pi}{A}\right)}{-\log(1+i)}\).

Key Concepts

Solving Equations

Solving an equation involves finding the value of one or more variables that makes the equation true. In our given exercise, the main goal is to isolate the variable \( n \). To do this, you need to perform operations that simplify the equation step-by-step.

Here, we start with the formula:

• Multiply both sides of the equation by the denominator to eliminate the fraction. This is a common technique to make equations easier to manage.
• Distribute any multiplied terms to further simplify the equation.
• Move terms that do not contain the variable \( n \) to the other side of the equation. This helps in isolating the term that includes \( n \).
• Finally, using operations such as division on both sides can fully isolate the variable.

Each step requires careful arithmetic manipulation so that the equation remains balanced. Solving equations often calls for these basic strategies to reach the solution.

Properties of Exponents

Understanding the properties of exponents is crucial when handling equations that contain power terms, like \((1+i)^{-n}\) in our problem.

Here are some key properties:

• Exponential terms like \((1+i)^{-n}\) are subject to specific rules. For instance, \(a^{-n} = \frac{1}{a^n}\).
• When you apply a power to a fraction, it applies to both the numerator and the denominator.
• Dividing powers with the same base involves subtracting the exponents.
• In our exercise, using the property \((a)^b = e^{b \cdot \ln(a)}\) helps convert the exponential into a form where a logarithm can be applied.

These properties not only simplify equations but are essential when functions like exponential growth or decay are involved. Applying them allows us to move from exponential to linear equations, which are often easier to manipulate.

Logarithms

Logarithms are the inverse operation of exponentiation, and they become instrumental when solving equations with exponents involved, such as in this exercise.

Key aspects of logarithms include:

• The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.
• For example, if \(b^y = x\), then \(\log_b(x) = y\).
• Logarithms transform multiplicative processes into additive ones, simplifying many equations.
• In this exercise, taking the logarithm of both sides allows us to handle the exponent \(n\) by turning it into a coefficient.

By using logarithms, the complicated equation becomes more manageable, transforming the power-related problem into simpler arithmetic or algebraic expressions that are easier to solve.

Isolation of Variables

Isolating a variable means to rearrange an equation so that the variable appears on one side of the equation and all other terms are on the opposite side. Let's see how this works in our example to find \( n \).

Important steps include:

• Identify and bring together all terms containing the variable on one side of the equation.
• Use algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable.
• In this exercise, we initially have components like \(-A(1+i)^{-n}\), and our goal is to rearrange those to \((1+i)^{-n} = \frac{A-Pi}{A}\).
• To further isolate \( n \), logarithms were applied to simplify and subsequently solve for \( n \).

This technique is critical in algebra because it allows you to explicitly solve for the variable that is of interest, making sense of complex mathematical relationships by breaking them down step by step. Through this process, isolating variables becomes a foundational skill in solving multi-step algebraic equations.