Problem 23 - Precalculus with Limits

Question

In Exercises 23 - 32, find a formula for for the arithmetic sequence. \( a_1 = 1, d = 3 \)

Step-by-Step Solution

Answer

The formula for the arithmetic sequence is \(a_n = 3n - 2\).

Step 1: Identifying provided variables

Given in the problem is the first term \(a_1 = 1\) and the common difference \(d = 3\).

Step 2: Applying the formula of arithmetic sequence

The general formula to find any term in an arithmetic sequence is \(a_n = a_1 + (n - 1) \cdot d\). Here, \(a_1 = 1\), \(d = 3\) are already provided.

Step 3: Substituting the values

Substituting the given values into the formula, we get: \(a_n = 1 + (n - 1) \cdot 3\). This simplifies to: \(a_n = 1 + 3n - 3\).

Step 4: Further simplifying the formula

Solving it will yield the equation \(a_n = 3n - 2\). This is the formula for given arithmetic sequence.

Key Concepts

First Term

In an arithmetic sequence, the first term is often represented by \(a_1\). The first term is the starting point from which the sequence progresses by repeatedly adding a fixed number. In our given example, the first term \(a_1 = 1\). This serves as the fundamental building block for forming the entire sequence. The first term is crucial because it sets the stage for all subsequent terms. Without it, the pattern of the sequence couldn't be established.
Remember, every arithmetic sequence requires this initial value, which starts the chain of the sequence. Always identify the first term when working with sequences to make sure calculations are accurate.

Common Difference

The common difference, denoted by \(d\), is a fixed number that you add to each term to get to the next term in an arithmetic sequence. In our exercise, the common difference is given as \(d = 3\). This means each term in the sequence is 3 greater than the term before it.
It is called 'common' because it remains the same throughout the sequence. Identifying the common difference is key to understanding the pattern and forming a formula for the sequence. If this number isn't consistent, the sequence would not be considered arithmetic.

General Formula for Arithmetic Sequence

To find any term in an arithmetic sequence, we use a general formula:

• \(a_n = a_1 + (n - 1) \cdot d\)

This formula enables us to calculate the value of any term given its position \(n\) without listing all previous terms. Let's break this down:

• \(a_n\) is the term you’re seeking.
• \(a_1\) is the first term.
• \(n\) represents the term's position.
• \(d\) is the common difference.

For example, substituting \(a_1 = 1\) and \(d = 3\) into the formula gives us \(a_n = 1 + (n-1) \cdot 3\), leading us to a more straightforward expression that matches our sequence.

Finding nth Term

To find the nth term in an arithmetic sequence, you use the formula:

• \(a_n = a_1 + (n - 1) \cdot d\)

For our specific sequence, we substitute the known values \(a_1 = 1\) and \(d = 3\) into the formula. Doing so, we find
\(a_n = 1 + (n-1) \cdot 3\). Simplifying further, it becomes \(a_n = 1 + 3n - 3 = 3n - 2\). This expression \(3n - 2\) represents the nth term of the sequence, giving us direct access to any term by simply knowing its position \(n\). For instance, to find the 5th term, just plug in \(n = 5\), resulting in \(a_5 = 3(5) - 2 = 13\). Therefore, mastering the formula simplifies finding any desired term in a sequence.