Problem 12 - 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?\(d .\) What is the 10 th term? $$a=\sqrt{3}, d=\sqrt{3}$$

Step-by-Step Solution

Answer

The 10th term is \(10\sqrt{3}\).

Step 1: Understand the Formula for the nth Term

The formula for the nth 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.

Step 2: Plug in the Known Values

We are given that the first term \(a = \sqrt{3}\) and the common difference \(d = \sqrt{3}\). We want to find the 10th term, so we set \(n = 10\).

Step 3: Calculate the 10th Term

Using the formula \(a_n = a + (n-1)d\), substitute the values: \(a_{10} = \sqrt{3} + (10-1)\sqrt{3}\).

Step 4: Simplify the Equation

Simplify \(a_{10} = \sqrt{3} + 9\sqrt{3}\) to get \(a_{10} = 10\sqrt{3}\).

Key Concepts

nth term formula

The concept of an "nth term" formula is central in understanding arithmetic sequences. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms, known as the "common difference," is constant.

The nth term formula for an arithmetic sequence is given by:

• \(a_n = a + (n-1)d\)

Here:

• \(a_n\) represents the nth term you want to find.
• \(a\) is the first term in the sequence.
• \(d\) is the common difference between consecutive terms.
• \(n\) is the term number you are interested in finding.

If you're trying to locate any term in such a sequence, this formula helps determine exactly what that term is by plugging in the values you know, such as the first term, the number of the term, and the common difference. Understanding this formula is essential because it provides a quick and efficient way to jump to any term directly without having to list the entire sequence.

common difference

In any arithmetic sequence, the "common difference" is a key factor that defines the pattern of change between terms. This difference, denoted as \(d\), is what separates each term from the next in the sequence.

To find the common difference:

• Subtract the first term from the second term of the sequence.

If you start with the sequence \(a, a+d, a+2d, ...\), the difference remains consistent between each successive term:

• \(d = (a+2d) - (a+d) = d\)

This consistency is crucial in creating what is termed an "arithmetic" pattern. In our case, the given common difference \(d = \sqrt{3}\) adds a steadily incrementing factor to each following term starting from the first term, \(a = \sqrt{3}\). Knowing the common difference allows you to predict all future terms if you know one term from the sequence.

precalculus concepts

Understanding arithmetic sequences is a fundamental precalculus concept that prepares you for more complex mathematical studies. It integrates basic principles of algebra and sequences to develop a foundational understanding of patterns and linear relationships.

In precalculus, arithmetic sequences are typically your first encounter with structured, predictable series of numbers. Mastery over these sequences gives you insight into:

• Pattern recognition, which is critical in solving mathematical problems.
• Understanding functions and their graphical representations.
• Building a basis for calculus, where limits and sequences play a significant role.

Beyond just arithmetic sequences, precalculus involves learning how to interpret and manipulate sequences and series, working with rational expressions, and managing diverse types of functions. The practice of identifying patterns, like those in arithmetic sequences, sharpens your analytical skills and prepares you for calculus challenges, such as solving rate of change problems and understanding the behavior of functions.