Problem 11 - Algebra for College Students

Question

List all terms of each finite sequence. \(c_{n}=(-2)^{n-1}\) for \(1 \leq n \leq 5\)

Step-by-Step Solution

Answer

The terms of the sequence are: 1, -2, 4, -8, 16.

Step 1: Understand the expression for the sequence

The sequence is given by the formula: \( c_{n} = (-2)^{n-1} \). This means for each value of from 1 to 5, plug it into the formula to find the corresponding term.

Step 2: Calculate the first term

For n = 1: \( c_{1} = (-2)^{1-1} = (-2)^{0} = 1 \)

Step 3: Calculate the second term

For n = 2: \( c_{2} = (-2)^{2-1} = (-2)^{1} = -2 \)

Step 4: Calculate the third term

For n = 3: \( c_{3} = (-2)^{3-1} = (-2)^{2} = 4 \)

Step 5: Calculate the fourth term

For n = 4: \( c_{4} = (-2)^{4-1} = (-2)^{3} = -8 \)

Step 6: Calculate the fifth term

For n = 5: \( c_{5} = (-2)^{5-1} = (-2)^{4} = 16 \)

Step 7: List all the terms

By compiling all calculated terms, the sequence is: \( 1, -2, 4, -8, 16 \)

Key Concepts

sequence calculation

When working with sequences, it is essential to understand how to calculate each term. Sequences are ordered lists of numbers following a specific rule. In this exercise, the given sequence is defined by the formula: \( c_{n} = (-2)^{n-1} \).
This type of problem requires you to substitute different values of \( n \) (ranging from 1 to 5) into the formula to find each term of the sequence. Let's break down the process:
First, identify the rule given by the sequence formula. For \( n = 1 \):
\( c_{1} = (-2)^{1-1} = (-2)^{0} = 1 \)
For \( n = 2 \):
\( c_{2} = (-2)^{2-1} = (-2)^{1} = -2 \)
Continue this process for \( n = 3 \), \( n = 4 \), and \( n = 5 \), you'll get the values 4, -8, and 16 respectively.
Once you've calculated all terms, you can list them out to form the complete sequence: 1, -2, 4, -8, 16.
By mastering sequence calculation, you can apply this method to various sequences defined by different rules.

exponential function

In this exercise, an exponential function describes the sequence. An exponential function is a mathematical expression where a constant base is raised to a variable exponent. Here, the base is -2, and the exponent is \( n-1 \).
The notation \( (-2)^{n-1} \) indicates repeated multiplication. Specifically, for any positive integer \( n \), the term is calculated by raising -2 to the power of \( n-1 \). This process can produce both positive and negative results, especially since raising a negative number to an odd power yields a negative outcome, while an even power results in a positive value.
Knowing the behavior of exponential functions makes it easier to predict and understand the pattern in sequences. For example:
- For \( n = 1 \), \( (-2)^{1-1} = 1 \) (positive).
- For \( n = 2 \), \( (-2)^{2-1} = -2 \) (negative).
- For \( n = 3 \), \( (-2)^{2} = 4 \) (positive).
The alternation between positive and negative values is a characteristic of this sequence.

mathematical terms

Understanding a few key mathematical terms is crucial for dealing with sequences and exponential functions:
1. **Term**: Each number in a sequence is called a 'term.' For example, 1, -2, 4, -8, and 16 are terms in this sequence.
2. **Exponent**: The number that indicates how many times the base is multiplied by itself. In \( (-2)^{n-1} \), \( n-1 \) is the exponent.
3. **Base**: The constant value that gets raised to the power of the exponent. Here, -2 is the base.
4. **Sequence**: An ordered list of numbers. Each number is associated with a position, called an index or subscript.
Proper comprehension of these terms helps decode and work through problems involving sequences and ensures a solid foundation for more complex mathematical concepts.
With these terms in mind, you can better grasp the nature of the sequence \( c_{n} = (-2)^{n-1} \) and effectively solve similar problems.