Problem 36 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(3,7,11,15, \dots\)

Step-by-Step Solution

Answer

The general term is: a_n = 4n - 1.

Step 1: Identify the first term

The first term of the sequence is given by the first number in the series. Here, the first term (denoted as 'a') is 3.

Step 2: Determine the common difference

To find the common difference (denoted as 'd'), subtract the first term from the second term or the second term from the third term. For this sequence: d = 7 - 3 = 4 (or) d = 11 - 7 = 4. So, the common difference 'd' is 4.

Step 3: Write the formula for the nth term

For an arithmetic sequence, the nth term is given by the formula a_n = a + (n - 1)d. Here, 'a' is the first term, and 'd' is the common difference. Substituting the values we have:a_n = 3 + (n - 1)4.

Step 4: Simplify the formula

Simplify the expression to get the final formula:a_n = 3 + 4(n - 1)a_n = 3 + 4n - 4a_n = 4n - 1.

Key Concepts

General Term Formula

To understand arithmetic sequences, we need to grasp how to find the general term formula. The general term formula for an arithmetic sequence is given by the expression: a_n = a + (n - 1)dHere,

• 'a' represents the first term of the sequence,
• 'd' is the common difference between the terms in the sequence, and
• 'n' represents the position term we are looking for.

For example, in the sequence 3, 7, 11, 15, ...,

• 'a' is 3,
• 'd' is 4.

Using the formula, the general term (a_n) can be written as:a_n = 3 + (n - 1)4Simplifying this, we get:a_n = 3 + 4n - 4a_n = 4n - 1So, the nth term of this sequence can be found using the expression 4n - 1. This formula enables us to find any term in the sequence by simply substituting the value of 'n'.

Common Difference

The common difference (denoted as 'd') is a key concept in arithmetic sequences. It is the amount by which consecutive terms increase or decrease. To find the common difference, you subtract the first term from the second term:d = 7 - 3 = 4Alternatively, you can subtract any term from the next term in the sequence:d = 11 - 7 = 4In our sequence 3, 7, 11, 15, ..., the common difference 'd' is consistently 4. This uniform increment is what makes the sequence arithmetic. Understanding the common difference is crucial because it helps us build the general term formula and analyze the sequence patterns.

Sequence Analysis

Analyzing an arithmetic sequence involves a few key steps:

• Identify the first term ('a')
• Determine the common difference ('d')
• Formulate the general term equation (a_n = a + (n - 1)d)

Let's apply these steps to our sequence (3, 7, 11, 15, ...). First, we identify 'a' as 3. Next, we find 'd' is 4. Then, we use these values to create the general term formula:a_n = 3 + (n - 1)4Simplifying yields:a_n = 4n - 1Using this formula, we can easily find any term in the sequence. For instance, to find the 5th term, we substitute 'n' with 5:a_5 = 4(5) - 1 = 20 - 1 = 19Thus, a_n = 4n - 1 allows us to predict all terms in the sequence and understand its behavior over time.