Problem 1 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{1}=200, d=20 $$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are 200, 220, 240, 260, 280, and 300.

Step 1: Identify the first term and common difference

Here, the first term, denoted by \(a_{1}\), is 200, and the common difference, represented by \(d\), is 20. In an arithmetic sequence, the difference between any two successive terms is a constant, which is \(d\) in this case.

Step 2: Calculation of the following terms

By adding the common difference to the first term, the second term of the sequence can be obtained. In this case, \(a_{2} = a_{1} + d = 200 + 20 = 220\). Then add the common difference to the second term to get the third term: \(a_{3} = a_{2} + d = 220 + 20 = 240\). Repeat this process to get the rest of the terms.

Step 3: Listing all six terms

By repeating the process in step 2, the first six terms are found to be 200, 220, 240, 260, 280, and 300.

Key Concepts

Common Difference in Arithmetic Sequence

Understanding the common difference in an arithmetic sequence is crucial for grasping how the sequence develops. An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This constant difference is known as the common difference, denoted by 'd'.

For instance, if you're given the sequence 3, 7, 11, 15, the common difference is 4, because each term is 4 units more than the previous term. Similarly, in our textbook exercise, the common difference is 20. By recognizing the common difference, we establish a pattern that allows us to predict subsequent terms of the sequence without having to list each one.

The concept of a common difference is not only foundational for arithmetic sequences, but also for understanding more complex mathematical concepts involving sequences and series.

Arithmetic Sequence Formula

The arithmetic sequence formula provides a quick and efficient way to find any term in an arithmetic sequence without having to add the common difference repeatedly. The formula for the nth term, denoted as \(a_n\), of an arithmetic sequence is given by:
\[a_n = a_1 + (n - 1)d\]
Where:\( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

For example, to find the 6th term in the sequence from our textbook problem, you would plug in the values of the first term (200) and the common difference (20) into the formula, giving us \( a_6 = 200 + (6 - 1) \times 20 = 300 \). This powerful formula empowers students to calculate the value of terms in lengthy sequences without performing extensive addition.

Sequence and Series

The concepts of sequence and series are fundamental in understanding various patterns and summations in mathematics. A sequence is an ordered list of numbers, where each number is called a term. On the other hand, a series is the sum of the terms of a sequence.

To distinguish, consider the arithmetic sequence from our problem: 200, 220, 240, 260, 280, 300. If we were to add these terms together, we would have a series. Series are often used in real-world applications such as calculating interest, analyzing patterns of growth or decline, and solving diverse mathematical problems.

It's valuable to recognize that while sequences list numbers and reveal patterns, series combine these numbers to give us a total sum. Mastery of these concepts fosters a deeper understanding of the intricacies of arithmetic and their practical applications.