Problem 21 - Beginning and Intermediate Algebra

Question

Given the arithmetic sequence \(4,7,10,13,16, \dots\) a) Find \(a_{1}\) and \(d\) b) Find a formula for the general term of the sequence, \(a_{n}\) c) Find the 35 the term of the sequence.

Step-by-Step Solution

Answer

The first term (\(a_{1}\)) is 4, the common difference (\(d\)) is 3, the formula for the general term of the sequence is \(a_{n} = 3n + 1\), and the 35th term of the sequence is 106.

Step 1: (Step 1: Identifying \(a_{1}\) and \(d\))

Since it is an arithmetic sequence, we know that the difference between consecutive terms is constant. Given the first few terms of the sequence: \(4, 7, 10, 13, 16, \dots\), we can deduce the following: 1. The first term (\(a_{1}\)) is simply the first term of the sequence, which is 4. 2. The common difference (\(d\)) can be found by subtracting the first term from the second term (or any consecutive terms): \(d = 7 - 4 = 3\). So, \(a_{1} = 4\) and \(d = 3\).

Step 2: (Step 2: Finding a formula for \(a_{n}\))

To find a formula for the general term of the sequence, \(a_{n}\), we use the arithmetic sequence formula: \[ a_{n} = a_{1} + (n - 1)d \] We already found \(a_{1} = 4\) and \(d = 3\). Therefore, the formula for \(a_{n}\) is: \[ a_{n} = 4 + (n - 1)3\] Now, we need to distribute the 3 and simplify the formula: \[ a_{n} = 4 + 3n - 3\] \[ a_{n} = 3n + 1 \] So the formula for the general term of the sequence is \(a_{n} = 3n + 1\).

Step 3: (Step 3: Finding the 35th term of the sequence)

Now with the general formula for \(a_{n} = 3n + 1\), we can find the 35th term of the sequence by substituting \(n = 35\): \[ a_{35} = 3(35) + 1 \] \[ a_{35} = 105 + 1 \] \[ a_{35} = 106 \] So the 35th term of the sequence is 106.

Key Concepts

Understanding Common Difference

In an arithmetic sequence, understanding the concept of a common difference is the key to unlocking its pattern. The common difference (\(d\)) is the fixed amount that you add to each term to get to the next in the sequence. You can find it by subtracting any term from the term that comes immediately after it.
For example, given the terms in an arithmetic sequence: 4, 7, 10, 13, 16, ... , you will notice a constant addition of 3 to each term.

• From 4 to 7, you add 3.
• From 7 to 10, another 3 is added.
• The pattern continues, maintaining the constant difference.

So, we can say the common difference for this sequence is 3.
The common difference helps in predicting any term in the series and is an essential part of forming the general term formula.

General Term Formula

An arithmetic sequence has a special formula that helps us calculate any term in the sequence without listing all other terms. This is known as the general term formula:\[ a_{n} = a_{1} + (n - 1)d \]Let's break it down:

• \(a_{n}\) = the nth term we're interested in finding.
• \(a_{1}\) = the first term of the sequence.
• \(d\) = common difference.
• \(n\) = the position of the term in the sequence.

For our example sequence 4, 7, 10, 13, 16, ..., we found that \(a_{1} = 4\) and \(d = 3\). So the general formula is developed as follows:
\[ a_{n} = 4 + (n - 1)3\]
Simplifying this yields:
\[ a_{n} = 3n + 1.\]
This formula allows for direct calculation of any term in the sequence, such as the 35th term without the need to list all previous terms.

Solving Arithmetic Sequence Problems

Solving problems related to arithmetic sequences often involves using the concepts of common difference and the general term formula together. Let's see how they help tackle specific arithmetic sequence problems, like finding a particular term or determining the sequence's pattern.
Say we are tasked with finding the 35th term of our sequence.

• First, identify the components: first term \(a_1\) and the common difference \(d\). For us, \(a_1 = 4\) and \(d = 3\).
• Next, apply the general term formula: \(a_{n} = 3n + 1\).
• Plug the desired term's position into the formula. For the 35th term (\(n = 35)\), substitute into the formula: \(a_{35} = 3(35) + 1\).
• Calculate to solve: \(a_{35} = 106\).

These steps illustrate how the systematic use of arithmetic sequence principles makes solving such problems straightforward.
You can answer similar questions simply by identifying the sequence's first term and common difference, then using the general term formula.