Problem 24 - College Algebra

Question

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$2,7,12,17, \dots$$

Step-by-Step Solution

Answer

The 20th term of the sequence is 97.

Step 1: Find the Common Difference

The common difference can be found by subtracting the first term from the second term, yielding \(d = 7 - 2 = 5.\)

Step 2: Write the General Formula for the nth Term

Now, using the formula for an arithmetic sequence \(a_{n} = a_{1} + (n-1) * d\), and the fact that \(a_{1} = 2\) and \(d = 5\), the general formula for the nth term is \(a_{n} = 2 + (n-1) * 5\). This can be simplified as \(a_{n} = 5n - 3\).

Step 3: Find the 20th Term

Substitute \(n = 20\) into the formula to find the 20th term: \(a_{20} = 5*20 - 3 = 97\).

Key Concepts

nth term formula

The nth term formula in an arithmetic sequence allows us to find any term in the sequence without having to list all the previous terms. This formula is derived from the characteristics of arithmetic sequences, which increase by a constant difference. In a sequence, if the first term is denoted by \( a_1 \) and the common difference by \( d \), the formula for the nth term, \( a_n \), is:

• \( a_n = a_1 + (n-1) \cdot d \)

Here:

• \( a_n \) represents the nth term of the sequence.
• \( a_1 \) is the first term.
• \( n \) is the position of the term in the sequence.
• \( d \) is the common difference.

For instance, in the sequence given in the exercise where the first term is 2 and the difference is 5, substituting these values gives \( a_n = 2 + (n-1) \cdot 5 \), which simplifies to \( a_n = 5n - 3 \). This formula is essential for calculating any term in the sequence efficiently.

common difference

The common difference in an arithmetic sequence is a crucial element as it defines how much each term increases or decreases to form the next term. It is represented by the letter \( d \). The common difference is found by subtracting any term in the sequence from the term that follows it.

• \( d = a_{n+1} - a_n \)

This constant helps maintain the uniform nature of an arithmetic sequence. For example, in the exercise, you can calculate it using the first two terms: \( 7 - 2 = 5 \). Therefore, the common difference \( d \) is 5. Ensuring you correctly identify this difference is vital as it directly affects the formula and finally the solution. The whole sequence is dependent on this common difference, making it a vital step in understanding how arithmetic sequences function.

arithmetic sequence problems

Arithmetic sequence problems often involve determining specific terms within the sequence or even the entire sequence itself using intuitive formulas. To solve these problems:

• Identify the first term, \( a_1 \).
• Determine the common difference, \( d \).
• Use the nth term formula, \( a_n = a_1 + (n-1) \cdot d \), to find the required term.
• For more complex problems, you might be asked to calculate sums or even invert the process to find \( n \) for a given term.

These sequences are foundational in math because they model how quantities change over equal intervals. For example, finding the 20th term in our sequence involves substituting \( n = 20 \) into the equation \( a_n = 5n - 3 \), yielding \( a_{20} = 5 \cdot 20 - 3 = 97 \). Practice will make navigating arithmetic sequence problems easier, helping in identifying patterns and creating formulas for quick solutions.