Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-(-5)^{n-1} $$

Step-by-Step Solution

Answer

The first four terms are -1, 5, -25, and 125.

Step 1: Understanding the question

We need to find the first four terms of the sequence given by the formula \( a_{n}=-(-5)^{n-1} \). This formula will be evaluated for \( n = 1 \), \( n = 2 \), \( n = 3 \), and \( n = 4 \).

Step 2: Calculate \( a_1 \)

Substitute \( n = 1 \) into the formula: \[ a_{1} = -(-5)^{1-1} = -(-5)^{0} = -1 \] So, the first term is \( a_1 = -1 \).

Step 3: Calculate \( a_2 \)

Substitute \( n = 2 \) into the formula: \[ a_{2} = -(-5)^{2-1} = -(-5)^{1} = 5 \] Thus, the second term is \( a_2 = 5 \).

Step 4: Calculate \( a_3 \)

Substitute \( n = 3 \) into the formula: \[ a_{3} = -(-5)^{3-1} = -(-5)^{2} = -25 \] Therefore, the third term is \( a_3 = -25 \).

Step 5: Calculate \( a_4 \)

Substitute \( n = 4 \) into the formula: \[ a_{4} = -(-5)^{4-1} = -(-5)^{3} = 125 \] Thus, the fourth term is \( a_4 = 125 \).

Key Concepts

Arithmetic Sequences

In algebra, an **arithmetic sequence** is a list of numbers with a constant difference between consecutive terms. It is often referred to as a linear sequence because each term increases or decreases by the same amount. This difference is known as the common difference.

The formula for an arithmetic sequence can be written as:

• \( a_n = a_1 + (n-1) \, d \)

Here, \( a_n \) is the \( n^{th} \) term, \( a_1 \) is the first term, and \( d \) represents the common difference. This equation allows you to find any term within an arithmetic sequence, given the first term and the common difference.

For example, if you have a sequence starting at 2 and increasing by 3 each time, it would look like 2, 5, 8, 11, etc. Each term is greater than the previous term by a difference of 3. Thus, understanding this concept helps identify patterns and find missing parts of a sequence.

Geometric Sequences

A **geometric sequence**, unlike an arithmetic sequence, involves a constant ratio between consecutive terms. Instead of a constant addition or subtraction, in a geometric sequence, each term is multiplied by the same number to get the next term. This number is called the common ratio.

The formula for finding the \( n^{th} \) term of a geometric sequence is:

• \( a_n = a_1 \cdot r^{(n-1)} \)

Here, \( a_n \) is the \( n^{th} \) term, \( a_1 \) is the first term, and \( r \) is the common ratio. By using this formula, you can easily determine any term in the sequence, provided you know the first term and the ratio.

For instance, in a geometric sequence where the first term is 3 and the common ratio is 2, the sequence would be 3, 6, 12, 24, etc. Each term is a result of multiplying the previous term by 2, showcasing the nature of geometric sequences.

Sequence Formula Interpretation

When interpreting a sequence formula, understanding how it generates the terms is crucial. Let's take the example of the exercise sequence provided: \( a_n = -(-5)^{n-1} \). This sequence might initially seem unclear, but careful examination reveals its nature.

First, note that the formula includes \((-5)^{n-1}\), which suggests a pattern similar to a geometric sequence since there is an exponent involved. Here, each term's sign alternates due to the presence of \(-1\) multiplication, and the terms are affected by powers of -5.

To find each term:

• Calculate \((-5)^{n-1}\).
• Apply the negative sign at the front of the term value.

By doing this, when \( n = 1 \), we calculate \(-(-5)^0\) resulting in \(-1\). For \( n = 2 \), we have \(-(-5)^1 = 5\), and so on for further terms.

This sequence formula highlights the importance of examining both the arithmetic and algebraic operations within a formula to fully interpret how the terms in the sequence develop.