Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=2^{n}-2 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are 0, 2, 6, and 14.

Step 1: Substitute n=1

To find the first term of the sequence, substitute \(n=1\) into the given formula \(a_n = 2^n - 2\). So, we get \(a_1 = 2^1 - 2 = 2 - 2 = 0\). Hence, the first term is 0.

Step 2: Substitute n=2

To find the second term, substitute \(n=2\) into \(a_n = 2^n - 2\). So, \(a_2 = 2^2 - 2 = 4 - 2 = 2\). Thus, the second term is 2.

Step 3: Substitute n=3

For the third term, use \(n=3\). Substitute this into the formula: \(a_3 = 2^3 - 2 = 8 - 2 = 6\). Therefore, the third term is 6.

Step 4: Substitute n=4

Finally, to find the fourth term, substitute \(n=4\) into the formula: \(a_4 = 2^4 - 2 = 16 - 2 = 14\). The fourth term is 14.

Key Concepts

Algebraic Expressions

Algebraic expressions form the basis of much of algebra and are a general way to represent mathematical ideas. They typically consist of numbers, variables, and operations. Variables, often denoted by letters such as \( n \), represent unknown or changeable values. When these components are combined with operations like addition, subtraction, multiplication, or division, they form an algebraic expression.

For example, in the given exercise, the algebraic expression \( a_n = 2^n - 2 \) is used to define the terms of a sequence. Here, \( 2^n \) represents an exponential term, and \(-2\) is a constant that adjusts the value after the power of two is calculated. This expression allows for the systematic calculation of sequence terms by simply substituting different values for \( n \).

Understanding how to interpret and manipulate algebraic expressions is crucial for solving many types of mathematical problems. It enables you to express relationships and patterns clearly and concisely.

Exponential Functions

Exponential functions are a class of mathematical functions where a constant base is raised to a variable exponent. They are commonly expressed in the form \( a^x \), where \( a \) is the base and \( x \) is the exponent.

In the exercise given, the expression \( 2^n \) is an exponential term because the base (2) is raised to the power of \( n \). This function grows rapidly as \( n \) increases. For instance:

• When \( n = 1 \), \( 2^1 = 2 \).
• When \( n = 2 \), \( 2^2 = 4 \).
• When \( n = 3 \), \( 2^3 = 8 \).
• When \( n = 4 \), \( 2^4 = 16 \).

Exponential functions are vital in many fields, including science and finance, because they describe growth patterns such as population expansion and compound interest.

Understanding how to work with exponential functions helps students see the rapid rates at which values can change, but it is essential to manage them properly, especially when substituting them into algebraic expressions.

Step by Step Problem Solving

Step by step problem solving is a methodical approach to breaking down a problem into manageable steps. It helps in understanding complex problems and finding solutions efficiently. This approach often involves

• Identifying what is being asked or needs to be solved.
• Understanding the given information and organizing it logically.
• Substituting known values into formulas where appropriate.
• Performing calculations step by step and verifying results.

In the exercise provided, a step by step strategy is employed to calculate the first four terms of the sequence defined by the expression \( a_n = 2^n - 2 \).

This involved systematically substituting the values \( n = 1, 2, 3, 4 \) into the expression and calculating the result for each to find the sequence terms 0, 2, 6, and 14, respectively. Working through problems systematically ensures clarity and minimizes errors, making it a powerful skill for solving a wide range of mathematical challenges.