Problem 10 - College Algebra

Question

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}+4, a_{1}=-7$$

Step-by-Step Solution

Answer

The first six terms of the sequence are -7, -3, 1, 5, 9, and 13.

Step 1: Identify the first term and the common difference

In this exercise, the first term \(a_1\) is -7 and the common difference is 4: each term is 4 greater than the preceding term.

Step 2: Apply the formula for the second term

The formula \(a_{n}=a_{n-1}+4\) can be used to find the second term. Substituting \(n = 2\) into the formula, the calculation becomes: \(a_{2}=a_{1}+4\). With \(a_1 = -7\), then \(a_2 = -7 + 4 = -3\)

Step 3: Apply the formula for the third term

Similarly, substituting \(n = 3\) yields: \(a_{3}=a_{2}+4\). With \(a_2 = -3\), it gives \(a_3 = -3 + 4 = 1\)

Step 4: Keep applying the formula until the sixth term

Continuing in this way for \(n = 4, 5, 6\) provides the succeeding terms. These are: \(a_{4} = 1 + 4 = 5\), \(a_{5} = 5 + 4 = 9\) and \(a_{6} = 9 + 4 = 13\)

Key Concepts

Common Difference

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the "common difference". In the given exercise, the common difference is 4. This means each term is found by adding 4 to the previous term.

Understanding the Importance

• The common difference allows you to predict future terms easily.
• It helps in creating a pattern that continues infinitely.
• This simple addition or subtraction helps to understand more complex topics later.

Calculating the common difference is straightforward. If you have two consecutive terms, subtract the first one from the second to find the difference. Even if it's a negative value, it remains consistent across the sequence.

Sequence Terms

Sequence terms refer to the individual numbers within a sequence. In an arithmetic sequence, each term is calculated based on the previous one with the help of the common difference.

Finding the Terms
For the exercise provided, the first term is given as \(a_1 = -7\). Each subsequent term is calculated by adding 4, the common difference, to the last known term.

To find more terms:

• The second term \(a_2 = a_1 + 4 = -3\).
• The third term \(a_3 = a_2 + 4 = 1\).
• Continue this process for more terms: \(a_4 = 5\), \(a_5 = 9\), and \(a_6 = 13\).

Each term builds on the previous, making calculations simple once you understand the pattern.

Recursive Formula

A recursive formula is a way of defining the terms of a sequence with respect to the previous terms. In the arithmetic sequence from the exercise, the recursive formula is \(a_{n} = a_{n-1} + 4\).

How It Works

• The formula starts with a base term, known as the first term \(a_1 = -7\).
• To find any other term, use the previous term \(a_{n-1}\) and add the common difference to it.
• This dependency on the previous term emphasizes understanding sequence flow and prediction of subsequent values.

A recursive approach provides deeper insights into sequences and helps manage complex problems by breaking them down into simpler steps.