Problem 41 - Precalculus Mathematics for Calculus

Question

Which term of the arithmetic sequence \(1,4,7, \ldots\) is \(88 ?\)

Step-by-Step Solution

Answer

The 30th term is 88.

Step 1: Identify the Sequence Pattern

This sequence is an arithmetic sequence where the first term is 1 and each subsequent term increases by 3. Thus, the common difference \(d\) is 3.

Step 2: Use the Formula for the n-th Term in an Arithmetic Sequence

The formula for the \(n\)-th term of an arithmetic sequence is given by: \(a_n = a + (n-1) \, d\), where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number we want to find.

Step 3: Substitute Known Values into the Formula

We know that the \(n\)-th term \(a_n = 88\), the first term \(a = 1\), and the common difference \(d = 3\). Substitute these values into the formula: \[88 = 1 + (n-1) \, 3\].

Step 4: Simplify and Solve for n

Simplify the equation: \[88 = 1 + 3n - 3\]. This simplifies to \[88 = 3n - 2\]. Adding 2 to both sides gives \[90 = 3n\]. Divide both sides by 3 to solve for \(n\): \[n = 30\].

Step 5: Conclusion

The 30th term of the sequence is 88.

Key Concepts

Common Difference

In an arithmetic sequence, each term is generated by adding a specific number, known as the "common difference," to the previous term. This value is constant throughout the sequence. For instance, in the sequence provided: 1, 4, 7, ..., the common difference is 3. Let's understand how this works in practice:

• The first term of the sequence is 1.
• The second term is calculated as 1 + 3 = 4.
• The third term follows as 4 + 3 = 7.
• This pattern continues through the sequence.

The common difference, often denoted by the symbol \( d \), plays a crucial role in determining the sequence pattern and calculating any term's position. Remember, the series will continue indefinitely unless a specific term is defined.

n-th Term Formula

To determine any term's position in an arithmetic sequence, we use the n-th term formula. This is a standardized formula, making it easier to find any term in the sequence without listing them all. The n-th term \( a_n \) is given by:\[ a_n = a + (n-1) \, d \] where:

• \( a_n \) is the term we aim to find.
• \( a \) is the first term of the sequence.
• \( n \) is the position of the term in the sequence.
• \( d \) is the common difference.

Let's break down how this works using our earlier example:- We want to find which term equals 88.- The first term \( a \) is 1, and the common difference \( d \) is 3.Hence, substituting in the formula gives: \[ 88 = 1 + (n-1) \times 3 \] Solving this helps us find that the term where \( a_n = 88 \) is indeed the 30th term.

Sequence Pattern

A sequence pattern is a predictable arrangement of numbers, typically based on arithmetic or geometric calculations. In arithmetic sequences, the pattern is linear, characterized by the constant addition of the common difference.
To detect a pattern in any arithmetic sequence, observe the following:

• Identify the start, or first term.
• Determine the regular interval added to reveal the common difference.
• Each subsequent number results from adding the common difference to the previous number.

With our sequence, 1, 4, 7, ..., it's evident that each number follows a clear growing pattern of adding 3. Recognizing patterns allows for quick calculations and predictions within sequences. For students, mastering sequence patterns simplifies comprehending complex sequence-related problems.