Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ 2,6,10,14, \ldots $$

Step-by-Step Solution

Answer

The nth term is \(4n - 2\), the fifth term is 18, and the tenth term is 38.

Step 1: Identify the First Term and Common Difference

In an arithmetic sequence, we need to first identify the first term, denoted as \(a_1\), and the common difference, denoted as \(d\). For the given sequence \(2, 6, 10, 14, \ldots \), the first term \(a_1 = 2\). The common difference \(d\) is calculated by subtracting the first term from the second term: \(d = 6 - 2 = 4\).

Step 2: Use the Formula for the nth Term

The formula for finding the \(n\)th term \(a_n\) of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). We'll use this formula to find the general formula for the \(n\)th term of the sequence.

Step 3: Formulate the nth Term

Substitute \(a_1 = 2\) and \(d = 4\) into the formula: \[a_n = 2 + (n-1) \cdot 4 = 2 + 4n - 4 = 4n - 2\]. So, the \(n\)th term of the sequence is \(a_n = 4n - 2\).

Step 4: Calculate the Fifth Term

To find the fifth term, substitute \(n = 5\) into the \(n\)th term formula:\[ a_5 = 4(5) - 2 = 20 - 2 = 18 \].Hence, the fifth term is 18.

Step 5: Calculate the Tenth Term

To find the tenth term, substitute \(n = 10\) into the \(n\)th term formula:\[ a_{10} = 4(10) - 2 = 40 - 2 = 38 \].Thus, the tenth term is 38.

Key Concepts

nth term formula

The nth term formula is a crucial tool in arithmetic sequences. It helps you find any term in the sequence quickly without having to write out all the previous terms. The formula is given by:

• \( a_n = a_1 + (n-1) \cdot d \)

In this formula:

• \( a_n \) is the nth term you're looking to find.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference between the terms.

For the sequence \(2, 6, 10, 14, \ldots\), the nth term formula becomes \( a_n = 4n - 2 \). This means that to find any term \( a_n \), you simply substitute \( n \) with the position number of the term you want to find, making the process efficient regardless of how large \( n \) is.
If you want to find the 50th or 100th term, no problem! Just plug \( n = 50 \) or \( n = 100 \) into the formula, and you'll get the term quickly without having to calculate each number leading up to it.

common difference

The common difference is another fundamental aspect of an arithmetic sequence. It is the amount that each term increases by as you move from one term to the next. To find the common difference:

• Subtract the first term from the second term.
• Alternatively, subtract any term from the term directly following it.

For the sequence \(2, 6, 10, 14, \ldots\), we find the common difference \( d \):

• \( d = 6 - 2 = 4 \)

The value of \( 4 \) means each term is 4 more than the term before it. Understanding the common difference allows prediction of future numbers in the sequence, without necessarily writing them all out. This regular increment is what makes an arithmetic sequence predictable and makes calculating terms straightforward.
Being clear about the common difference is vital in building confidence with sequences, as it is reference for the step-by-step progression of the terms.

finding terms of a sequence

Now that we have the necessary tools, finding specific terms in an arithmetic sequence is straightforward. We can use the nth term formula for accuracy and speed.To find any term, do the following:

• Identify \( n \), the position number of the term you wish to find.
• Use the formula \( a_n = a_1 + (n-1) \cdot d \).
• Substitute \( n \) and simplify.

For practice, let's find some terms in our sequence:

• **Fifth term (\( n = 5 \)):**\[ a_5 = 4 \cdot 5 - 2 = 18 \]
• **Tenth term (\( n = 10 \)):**\[ a_{10} = 4 \cdot 10 - 2 = 38 \]

This method not only gives you the answer, but it quickly builds confidence in working with any arithmetic sequence. It's all about plugging in and solving using what you already know about the sequence's structure. Once you master these steps, finding any term in an arithmetic sequence becomes intuitive and efficient.