Problem 16 - 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? $$a=\sqrt{3}, \quad d=\sqrt{3}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the formula for the n-th term

The n-th term of an arithmetic sequence can be found using the formula \(a_n = a + (n - 1) \cdot d\), where \(a\) is the first term and \(d\) is the common difference.

Step 2: Substitute given values into the formula

Substitute \(a = \sqrt{3}\) and \(d = \sqrt{3}\) into the formula for the n-th term to get \(a_n = \sqrt{3} + (n - 1) \cdot \sqrt{3}\).

Step 3: Simplify the expression

By factoring out \(\sqrt{3}\), the expression becomes \(a_n = \sqrt{3} \cdot (1 + (n - 1)) = \sqrt{3} \cdot n\).

Step 4: Find the 10th term

Use the simplified expression to find the 10th term by substituting \(n = 10\), giving us \(a_{10} = \sqrt{3} \cdot 10 = 10\sqrt{3}\).

Key Concepts

Nth Term

In an arithmetic sequence, the nth term is an important concept because it helps you find any term in the sequence without having to list out all the terms before it. The formula to find the nth term is given by:

• \( a_n = a + (n - 1) \cdot d \)

Here, \( a \) represents the first term of the sequence, and \( d \) stands for the common difference between consecutive terms. This formula makes it easy to directly calculate any term in the sequence when you know these two values.
You simply choose the position of the term you want to find by setting \( n \) to that number. For example, if you're looking for the 10th term, \( n \) would be 10. This feature of arithmetic sequences allows you to quickly and efficiently calculate terms without needing to perform each individual addition repeatedly.

First Term

The first term of an arithmetic sequence is the starting point from which the sequence begins. In the given exercise, the first term is \( a = \sqrt{3} \). This term sets the initial value for generating the rest of the sequence through repeated addition of the common difference.
To calculate any other term in the sequence, you begin with this first term and then apply the formula for the nth term. Remember that knowing this first term is crucial because it anchors the sequence and ensures all other terms can be accurately determined. Without this first number, determining subsequent terms within the sequence becomes impossible. This foundational element of arithmetic sequences significantly impacts the entire calculation process.

Common Difference

The common difference, denoted by \( d \), is the constant amount that each term in an arithmetic sequence differs from the previous one. For the exercise, the common difference is \( d = \sqrt{3} \). It is consistently added to each term to move to the next one. For instance, if you start with a term of \( a = \sqrt{3} \), the next term will be \( a + d = 2\sqrt{3} \), the subsequent term would be \( 3\sqrt{3} \), and so on.
This uniformity in difference is what defines an arithmetic sequence and differentiates it from other types of sequences. Understanding the role of the common difference helps you not only find subsequent terms efficiently but also reveals the linear nature of arithmetic sequences. It's the steady, predictable change between terms that gives these sequences their "arithmetic" quality. By applying this concept, you can map out an entire sequence once the first term and common difference are known.