Problem 49 - Precalculus with Limits

Question

In Exercises 47 - 50, the first two terms of the arithmetic sequence are given. Find the missing term. \( a_1 = 4.2, a_2 = 6.6, a_7 = \)

Step-by-Step Solution

Answer

The missing term, \( a_7 \), is 18.6.

Step 1: Find the common difference

In an arithmetic sequence, the difference between two subsequent terms is always constant and can be found by subtracting the first term from the second one. In this scenario, the common difference \(d\) can be found as: \( d = a_2 - a_1 = 6.6 - 4.2 = 2.4 \)

Step 2: Use the formula for the nth term in an arithmetic sequence

The formula for the nth term in an arithmetic sequence is: \( a_n = a_1 + (n-1) * d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step 3: Solve for the 7th term

Insert \( n = 7, a_1 = 4.2, d = 2.4 \) into the formula: \( a_7 = 4.2 + (7-1) * 2.4 = 18.6 \)

Key Concepts

Common Difference in Arithmetic Sequences

The common difference is what makes arithmetic sequences consistent and predictable. It is the fixed amount that each term increases or decreases by as you move from one term to the next. In any arithmetic sequence, you can calculate this by subtracting the first term from the second. For example, if your first term is 4.2 and the second term is 6.6, the difference is 2.4. This number is consistently added to each term to obtain the next term, ensuring the sequence progresses smoothly. Understanding the common difference is essential as it is one of the core components determining the structure of the entire sequence.

Understanding the nth Term Formula

The nth term formula is a powerful tool when working with arithmetic sequences. This formula is expressed as \( a_n = a_1 + (n-1) \times d \). Here, \( a_n \) represents the term we are interested in, \( a_1 \) is the first term of the sequence, \( n \) is the position of the term within the sequence, and \( d \) is the common difference.

This formula allows you to find any term in the sequence without having to calculate each preceding term. By inputting the known values, such as \( a_1 = 4.2 \), \( n = 7 \), and \( d = 2.4 \), you can efficiently solve for the 7th term or any other position within the sequence. This makes it a vital skill when solving sequences in math.

Exploring Sequence Terms

Sequence terms are the individual elements that make up the arithmetic sequence. Each term can be identified by its position or order in the sequence, starting from the first term and moving onwards. When a sequence is governed by a consistent common difference, each term builds upon the previous one by adding this difference.

For example, in a sequence starting with 4.2 and having a common difference of 2.4, the terms would be 4.2, 6.6, and so on. The sequence term's position is crucial when applying the nth term formula, as it determines the number of times the common difference is added to the initial term. This relationship helps to easily find missing terms in any arithmetic sequence, such as deducing the 7th term, which in our example would be 18.6.