Problem 29 - Precalculus with Limits

Question

In Exercises 23 - 32, find a formula for for the arithmetic sequence. \( a_1 = 5, a_4 = 15 \)

Step-by-Step Solution

Answer

The formula for the arithmetic sequence is \(a_n = 5 + (n-1) * 3.33\).

Step 1: Find the Common Difference

The common difference (\(d\)) can be found by subtracting the first term from the fourth term and dividing by the difference in their indices, which is 3 in this case: \[d = \frac{a_4 - a_1}{4-1} = \frac{15 - 5}{3} = \frac{10}{3} = 3.33\].

Step 2: Construct the Formula

The formula for an arithmetic sequence is: \[a_n = a_1 + (n-1) * d\]. Plugging in the values we have found, we get: \[a_n = 5 + (n-1) * 3.33\]

Key Concepts

Common Difference

The concept of the 'common difference' lies at the very heart of what defines an arithmetic sequence. In such sequences, each term after the first is created by adding a constant amount to the previous term. This fixed amount is what we refer to as the common difference, often denoted in textbooks and equations as 'd'.

To illustrate the importance of the common difference, consider our example where the first term of the sequence (\(a_1\)) is 5 and the fourth term (\(a_4\)) is 15. To determine the common difference, we subtract the first term from the fourth term and divide by the number of terms between them, which is 3 in this case. It gives us a common difference of \(d = \frac{a_4 - a_1}{4-1} = \frac{10}{3} = 3.33\). Once we know the common difference, we can use it to find any term in the sequence. The use of the common difference is indispensable because it acts as the 'stepping stone' that bridges each term in the sequence.

Sequence Indices

Sequence indices are equally as important in understanding arithmetic sequences as the common difference. They tell us the position of each term in the sequence. In our context, the index is represented by 'n', and it serves as a placeholder for the number of terms.Indices give us a way to refer to the terms in a sequence without always having to write them out. For instance, in our exercise, \(a_1\) refers to the first term, which has an index of 1, while \(a_4\) is the fourth term, with an index of 4. The formula to find any term in the sequence, \a_n = a_1 + (n-1) * d\, uses the sequence indices as an integral component, allowing us to substitute in the value of 'n' to find the nth term. Thus, understanding indices and how to manipulate them is crucial for properly applying the arithmetic sequence formula.

Arithmetic Sequence Terms

When we discuss 'arithmetic sequence terms', we're referring to the individual elements that make up the sequence itself. Each term is part of a larger pattern defined by the starting term, known as the first term (\(a_1\)), and the common difference ('d').

In our exercise, the task is to construct the general formula for terms of the arithmetic sequence. We identified the first term (5) and calculated the common difference (3.33), which then allows us to express the nth term (\(a_n\)) as a function of these two core components in the formula \[a_n = a_1 + (n-1) * d\]. By doing this, we can find any term in the sequence, regardless of its position (index). Each arithmetic sequence term can, therefore, be seen as an offshoot of the sequence's defining properties: its initial value and the increment by which it grows.