Problem 24 - Precalculus with Limits

Question

In Exercises 23 - 32, find a formula for for the arithmetic sequence. \( a_1 = 15, d = 4 \)

Step-by-Step Solution

Answer

The formula for the arithmetic sequence is \( a_n = 4n + 11 \).

Step 1: Identify the given values

The first term \( a_1 \) is given as 15 and the common difference \( d \) is given as 4.

Step 2: Substitute the values into the formula

Replace \( a_1 \) and \( d \) in the formula \( a_n = a_1 + (n-1) * d \) with the given values. This leads to \( a_n = 15 + (n-1) * 4 \).

Step 3: Simplify the formula

Simplify the expression to get the final formula for the arithmetic sequence. The simplified formula is \( a_n = 15 + 4n - 4 \), which further simplifies to \( a_n = 4n + 11 \).

Key Concepts

Sequence Formula

An arithmetic sequence is a list of numbers where each term is derived by adding a constant value to the previous term. This constant is known as the common difference. The general formula for finding any term in an arithmetic sequence is given by:
\[ a_n = a_1 + (n-1) \cdot d \]
Where:

• \( a_n \) is the \( n^{th} \) term you're trying to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the position of the term in the sequence.
• \( d \) is the common difference.

Using this formula allows you to calculate any term in the sequence without needing to write down all previous terms. This is especially useful when you need to find terms far along in the sequence, saving you time and effort. Remember that simplifying the formula will often make calculations easier and more straightforward.

Common Difference

The common difference is a crucial concept in any arithmetic sequence. It's the number you add to each term to get the next term. In the formula \( a_n = a_1 + (n-1) \cdot d \), the common difference is represented by \( d \). To identify the common difference:

• Subtract any term from the term that follows it in the sequence.
• This difference should remain constant throughout the sequence.

In our example exercise, the common difference \( d \) is 4. This means each term in the sequence increases by 4 from the previous one. Understanding and identifying the common difference allows you to discern the pattern within the sequence and is essential for constructing the sequence formula.

First Term

The first term of an arithmetic sequence, denoted as \( a_1 \), is the starting point of the sequence. It is the initial value from which all subsequent terms are derived by repeatedly adding the common difference, \( d \). For example, in the given exercise, the first term \( a_1 \) is 15.
This first term anchors the entire sequence and provides a reference point. Knowing the first term enables you to apply the sequence formula accurately:

• Simply plug \( a_1 \) into the formula \( a_n = a_1 + (n-1) \cdot d \).
• This gives you the desired term's value based on its position.

Thus, the first term not only establishes the beginning of the sequence but also serves as a key component in the calculation of any other term within the sequence.