Problem 9 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}+6, a_{1}=-9 $$

Step-by-Step Solution

Answer

The first six terms of the sequence are -9, -3, 3, 9, 15, 21.

Step 1: Identify the first term and common difference

In the given arithmetic sequence, the first term \(a_1\) is -9 and the common difference \(d\) is 6.

Step 2: Calculate the second term

Apply the arithmetic sequence formula which is \(a_n = a_{n-1} + d\) to find the second term. Here, \(n=2\), \(a_{n-1}=a_1=-9\), and \(d=6\). Therefore, \(a_2=a_{n-1}+d=a_1+d=-9+6=-3\). So the second term is -3.

Step 3: Calculate the third term

Using the same formula \(a_n = a_{n-1} + d\), for the third term, \(n=3\), \(a_{n-1}=a_2=-3\), and \(d=6\). Therefore, \(a_3=a_{n-1}+d=a_2+d=-3+6=3\). So the third term is 3.

Step 4: Calculate the fourth, fifth and sixth term

Repeating the same process, we obtain the fourth, fifth and six terms as follows: \nFourth term \(a_4=a_{n-1}+d=a_3+d=3+6=9\). \nFifth term \(a_5=a_{n-1}+d=a_4+d=9+6=15\). \nSixth term \(a_6=a_{n-1}+d=a_5+d=15+6=21\).

Key Concepts

Common Difference in Arithmetic Sequence

Understanding the term 'common difference' is crucial when exploring arithmetic sequences. It is simply the constant amount that each term in an arithmetic sequence changes from the previous one. It can be positive or negative, leading to an increasing or decreasing sequence, respectively.

For example, if you have a sequence where each term is 6 more than the one before it, your common difference is 6. In the exercise given, the common difference is stated as 6, which means to find each subsequent term, you add 6 to the previous term. It's like taking regular steps, where each step is of the same length - that's how you can visualize the role of the common difference in an arithmetic sequence.

Arithmetic Sequence Formula

The arithmetic sequence formula allows you to find any term within an arithmetic sequence without having to list all preceding terms. This formula is typically expressed as \(a_n = a_1 + (n - 1)d\), where \(a_n\) is the nth term you're looking for, \(a_1\) is the first term of the sequence, \(n\) is the position of the term, and \(d\) is the common difference.

Using this formula, you can directly calculate any term in the sequence. As shown in the exercise, to calculate the second term, you plug the values into the formula and get \(a_2 = a_1 + d = -9 + 6 = -3\). This formula streamlines the process significantly and is a fundamental tool when working with arithmetic sequences.

Calculating Terms in an Arithmetic Sequence

Calculating terms in an arithmetic sequence is straightforward once you comprehend the arithmetic sequence formula. After identifying the first term and common difference, you apply the formula \(a_n = a_{n-1} + d\) to determine the following terms. As you can see in the given exercise, for each new term, you take the previous term and add the common difference.

For instance, once you know the second term is -3, finding the third term involves adding the common difference of 6 again, resulting in 3. The process continues in this manner to compute subsequent terms, showing a clear pattern that can be easily followed. This exercise demonstrates an excellent use of the arithmetic sequence formula in a practical, step-by-step manner, emphasizing the underlying logic and consistency inherent in arithmetic sequences.