Problem 15 - Prealgebra

Question

Describe each sequence using words and symbols. $$8,16,24,32, \dots$$

Step-by-Step Solution

Answer

The sequence is defined by the formula \( a_n = 8n \), where \( n \) is the term's position.

Step 1: Identify the Pattern

First, look at how the sequence changes from one term to the next. Here, each term increases by 8 from the previous term. This indicates that the sequence is an arithmetic sequence.

Step 2: Define the Common Difference

In an arithmetic sequence, the common difference is the amount being added to each term to get the next term. For this sequence, the common difference is 8. This can be written as: \( d = 8 \).

Step 3: Define the First Term

The first term of the sequence is the starting value, which is 8. This is represented by \( a_1 = 8 \).

Step 4: Write the General Formula

The formula for the nth term of an arithmetic sequence is given by: \( a_n = a_1 + (n-1) \cdot d \).Substituting in the values, it becomes: \( a_n = 8 + (n-1) \cdot 8 \).

Step 5: Simplify the General Formula

Simplify the expression for the nth term:\( a_n = 8 + 8(n-1) \)\( a_n = 8 + 8n - 8 \)\( a_n = 8n \).Thus, each term in the sequence can be described as eight times the term’s position number.

Key Concepts

Common Difference

In any arithmetic sequence, the change between consecutive terms is constant. This change is called the "common difference." It is a fundamental characteristic that sets arithmetic sequences apart from other types of sequences. To determine the common difference, simply subtract any term in the sequence from the term that follows it. For instance, in the sequence given by the problem, the terms are 8, 16, 24, 32, and so on. Subtracting the first term from the second gives:

• \( 16 - 8 = 8 \)

Repeating this with the next pair of terms (24 - 16) or (32 - 24), you will continue to obtain 8, confirming that our common difference, \( d \), is consistently 8. This steady increase means each term is exactly 8 more than the previous one. When working with arithmetic sequences, identifying the common difference is essential as it helps in establishing a pattern throughout the sequence.

First Term

Understanding the starting point of an arithmetic sequence is crucial. This is known as the "first term," and is usually denoted as \( a_1 \). In the sequence you are exploring, the first term is 8. The significance of the first term lies in its foundational role. Every subsequent term in the sequence can be mapped back to this first term by adding multiples of the common difference.

• For example, beginning with 8, you then add the common difference, 8, to reach 16.
• Repeat this process, and you'll find every number in the sequence stems from the first term plus a product involving the common difference.

Identifying the first term provides a base from which the rest of the sequence can be explored or constructed. It also plays a vital role in the calculation of any term using the nth term formula.

Nth Term Formula

To find out any particular term in the sequence, you use the "nth term formula." This formula captures the entire sequence in a single expression and allows you to calculate the value of any term you want. For an arithmetic sequence, the nth term formula is given by:\( a_n = a_1 + (n-1) \cdot d \)In this formula:

• \( a_n \) represents the nth term you are solving for.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference.

Using the specific values from the given sequence:

• \( a_1 = 8 \)
• \( d = 8 \)

Substitute these into the formula to get:\( a_n = 8 + (n-1)\cdot 8 \)When simplified, this becomes \( a_n = 8n \). This equation means that the nth term is simply 8 times its position in the sequence. By understanding and applying this formula, you can leap directly to the nth term without having to list out all preceding terms in the sequence.