Problem 47 - Precalculus with Limits

Question

In Exercises 47 - 50, the first two terms of the arithmetic sequence are given. Find the missing term. \( a_1 = 5, a_2 = 11, a_{10} = \)

Step-by-Step Solution

Answer

The missing term \( a_{10} \) is 59.

Step 1: Identify the common difference

For an arithmetic sequence, the common difference \( d \) is obtained by subtracting the first term from the second term. In this case, \( d = a_2 - a_1 = 11 - 5 = 6 \). So, the common difference is 6.

Step 2: Apply the arithmetic sequence formula

The formula for finding \( a_n \) in an arithmetic sequence is \( a_n = a_1 + (n - 1)d \). Replace \( n \) with 10, \( a_1 \) with 5 (the first term), and \( d \) with 6 (the common difference) to find the 10th term. This gives \( a_{10} = 5 + (10 - 1)6 = 5 + 54 = 59 \).

Key Concepts

Common Difference

The common difference is a key concept in any arithmetic sequence. It represents the consistent interval between consecutive terms in the sequence. Understanding the common difference is crucial for identifying the pattern of the sequence and for predicting future terms.

Finding the Common Difference

To find the common difference, simply subtract the first term from the second term of the sequence. For example, given the first two terms of an arithmetic sequence, such as in the exercise, where the first term (\( a_1 \) is 5 and the second term (\( a_2 \) is 11, the common difference (\( d \) is calculated as:\br>\br> \[ d = a_2 - a_1 = 11 - 5 = 6 \]
Once the common difference is identified, it becomes the building block for generating other terms of the sequence and is essential for using the arithmetic sequence formula effectively.

Arithmetic Sequence Formula

The arithmetic sequence formula allows you to find any term in an arithmetic sequence. The formula is given by:\br>\[ a_n = a_1 + (n - 1)d \]
Here, \( a_n \) represents the term you wish to find, \( a_1 \) is the first term of the sequence, \( n \) is the position of the term in the sequence, and \( d \) is the common difference.

Applying the Formula

It's important to substitute the correct values into the formula to find your desired term. If the exercise asks for the 10th term of the sequence where the first term is 5 and the common difference is 6, you plug the values into the formula like so:\br>\br>\[ a_{10} = 5 + (10 - 1)\cdot6 \]
\br>This calculation leads to the term at position 10 being 59. Mastering this formula enables students to navigate through the sequence smoothly and to solve problems accurately and efficiently.

Sequence Terms

The terms of an arithmetic sequence are the individual elements that make up the sequence. Each term in the sequence has a unique position identified by its ordinal number, often denoted by \( n \), and is derived based on the preceding term and the common difference.

Understanding Sequence Terms

The first term, known as \( a_1 \) is the starting point for the sequence. From there, each succeeding term is the sum of the previous term and the common difference. This pattern continues systematically throughout the sequence. In our example, starting with \( a_1 = 5 \) and a common difference of 6, the terms unfold as follows:\br>\br>5 (first term), 11 (second term), 17 (third term), and so on.
\br>Being able to identify and articulate each term in the sequence is fundamental to understanding arithmetic sequences as a whole. Moreover, it sets the stage for grasping more advanced concepts in algebra and beyond.