Problem 8 - Algebra and Trigonometry

Question

5-8 Find the \(n\)th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10th term? $$a=\sqrt{3}, d=\sqrt{3}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the problem

We are asked to find the 10th term of an arithmetic sequence given the first term \(a\) and the common difference \(d\). The first term \(a\) is \(\sqrt{3}\) and the common difference \(d\) is also \(\sqrt{3}\).

Step 2: Recalling the formula for the nth term

In an arithmetic sequence, the \(n\)th term \(a_n\) can be calculated using the formula: \[a_n = a + (n-1) \cdot d\]. Here, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number we want to find.

Step 3: Applying the formula to find the 10th term

We substitute \(a = \sqrt{3}\), \(d = \sqrt{3}\), and \(n = 10\) into the formula for the \(n\)th term: \[a_{10} = \sqrt{3} + (10-1) \cdot \sqrt{3}\].

Step 4: Simplifying the expression

Calculate \(10 - 1 = 9\) and multiply by \(\sqrt{3}\), which gives \(9 \cdot \sqrt{3}\). The expression becomes: \[a_{10} = \sqrt{3} + 9\sqrt{3} = 10\sqrt{3}\].

Key Concepts

nth term formula

To find any specific term in an arithmetic sequence, we use the "nth term formula." This formula allows us to determine the value of any term by using the first term of the sequence and the common difference. The mathematical expression for the nth term is:

• \( a_n = a + (n-1) \cdot d \)

where:

• \( a_n \) represents the nth term we want to find,
• \( a \) is the first term of the sequence,
• \( d \) is the common difference between consecutive terms,
• and \( n \) is the number of the term in the sequence.

This formula is extremely useful because it provides a straightforward way to find the value of any term without having to add the common difference repeatedly. For example, if we want to find the 10th term in a sequence where the first term \( a \) is \( \sqrt{3} \) and the common difference \( d \) is also \( \sqrt{3} \), we would plug these values into the formula as follows:\[ a_{10} = \sqrt{3} + (10-1) \cdot \sqrt{3} \].Following this, you can quickly calculate the specific term needed.

common difference

The common difference is a critical element in understanding and working with arithmetic sequences. It is the amount by which each term in the sequence increases (or decreases) from the previous term. The common difference is consistent throughout the entire sequence.In our example, the common difference, \( d \), is \( \sqrt{3} \). This means that from the first term forward, each subsequent term is obtained by adding \( \sqrt{3} \) to the previous term.

• If the sequence is \( a_1, a_2, a_3, \ldots \), then \( a_2 = a_1 + d \), \( a_3 = a_2 + d \), and so on.

Knowing the common difference helps in forming a pattern or structure within the sequence, which is pivotal for calculating any further terms.

first term

The first term of an arithmetic sequence is the starting point of the sequence. It is often denoted by \( a \) and is crucial because it serves as the baseline from which all subsequent terms are derived. For example, in our exercise, the first term \( a \) is \( \sqrt{3} \).

• This first term is the reference for determining any other term in the sequence using the nth term formula.
• In a sequence such as our example, the first term is used to calculate the 10th term: starting with \( \sqrt{3} \) and adding the common difference \( \sqrt{3} \) nine more times since \( a_{n} = a + (n-1) \cdot d \).

The first term must be known to apply the nth term formula correctly and to ensure accuracy in further calculations.