Problem 11 - Precalculus: Mathematics for Calculus

Question

Find the \(n\) th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10 th term? $$a=9, \quad d=4$$

Step-by-Step Solution

Answer

The 10th term is 45.

Step 1: Understand the formula for the nth term

The formula for finding the nth term of an arithmetic sequence is given by \( a_n = a + (n-1) imes d \), where \(a_n\) is the nth term, \(a\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

Step 2: Identify the given values

From the problem, we have the first term \(a = 9\) and the common difference \(d = 4\). We need to find the 10th term, so \(n = 10\).

Step 3: Substitute the values into the formula

Substitute the known values into the arithmetic sequence formula: \[ a_n = 9 + (10-1) \times 4 \].

Step 4: Calculate the result

Simplify the expression: \( a_n = 9 + 9 \times 4 = 9 + 36 = 45 \).

Step 5: State the 10th term

The 10th term of the arithmetic sequence is 45.

Key Concepts

Understanding the nth term formula

The nth term formula helps you find any specific term in an arithmetic sequence without listing all the terms. An arithmetic sequence is a list of numbers with a fixed, constant difference between them. This difference is known as the common difference. Understanding the formula is crucial for solving many math problems related to sequences.
The formula for the nth term is given by:

• \( a_n = a + (n-1) \times d \)

Here:

• \(a_n\) represents the nth term you want to find.
• \(a\) is the first term of the sequence.
• \(n\) is the position of the term within the sequence.
• \(d\) is the common difference.

For example, if you start with a first term of 9 and a common difference of 4, you can substitute these values into the formula to find any term without listing them all.

The role of common difference

The common difference is a fundamental part of any arithmetic sequence. It is the constant amount that each term increases (or decreases, if negative) from the previous term. This makes it easy to recognize and work with arithmetic sequences.
To identify the common difference, simply subtract any term from the subsequent term in the sequence. For example, if the sequence starts with 9, 13, 17, ..., the common difference \(d\) is:

• \(d = 13 - 9 = 4\)

This tells us that each number is 4 units higher than the one before it. Knowing this pattern helps predict and calculate future terms accurately.

Calculating a specific term

Once you understand the nth term formula and common difference, calculating any specific term becomes a straightforward process. Let’s see how this works with an example.
Suppose you want to find the 10th term in a sequence where the first term \(a\) is 9 and the common difference \(d\) is 4. You already know \(n = 10\) because you are finding the 10th term.
Here's how you calculate it:

• Start with the nth term formula: \(a_n = a + (n-1) \times {d}\).
• Substitute the given values: \(a_{10} = 9 + (10-1) \times 4\).
• Perform the calculation by first solving \(10-1 = 9\), then \(9 \times 4 = 36\).
• Add 36 to the first term, 9, to get 45: \( a_{10} = 45 \).

The 10th term in this sequence is 45, following the pattern given by the first term and common difference.