Question

Are the sequences in Exercises \(1-4\) arithmetic? $$ 3,8,13,18, \ldots $$

Short answer

Answer: Yes, the sequence is arithmetic.

Step-by-step solution

Write down the given sequence

First, let's write the given sequence: $$ 3, 8, 13, 18, \ldots $$

Calculate the differences between consecutive terms

Now, find the difference between consecutive terms in the sequence: $$ 8 - 3 = d_1 \\ 13 - 8 = d_2 \\ 18 - 13 = d_3 $$

Verify if the differences are equal

Check if \(d_1\), \(d_2\), and \(d_3\) are equal: $$ d_1 = 5 \\ d_2 = 5 \\ d_3 = 5 $$ All the differences are equal to each other.

Determine if the sequence is arithmetic

Since all the differences are equal (\(d_1 = d_2 = d_3 = 5\)), we can conclude that the given sequence is arithmetic.

Key concepts

Difference between Terms

When dealing with sequences, one significant step is finding the difference between consecutive terms. This difference is crucial because it helps determine the nature of the sequence. In our example, the sequence given is: 3, 8, 13, 18, ...

To find the difference, you subtract the first term from the second term, the second term from the third, and so on. This step can be summarized as follows:

• First difference: 8 - 3 = 5
• Second difference: 13 - 8 = 5
• Third difference: 18 - 13 = 5

If all calculated differences are the same, we can conclude that the sequence has a consistent difference, sometimes called the "common difference." In this case, it's 5. Such consistency indicates a specific type of sequence, which brings us to our next topic.

Sequences

Sequences are ordered lists of numbers following a specific pattern or rule. Each of these numbers is known as a "term."

In our sequence example, we've seen numbers like 3, 8, 13, 18, etc. These numbers appear in a particular order, and each subsequent number is derived based on a rule using previous terms. The rule could be adding a fixed number (like in our example), multiplying, or even using a more complex relation.

Understanding sequences is essential because they appear in many areas of mathematics and everyday life. They are foundational in topics such as series, calculus, and even computer science. Recognizing the rule or pattern governing a sequence is the first step in solving problems related to sequences.

Arithmetic Progression

An arithmetic sequence is also known as an arithmetic progression. It's defined as a sequence where the difference between any two consecutive terms is constant. This means that once a common difference is identified between the terms of a sequence, it is consistently applied throughout the sequence.

In our given sequence 3, 8, 13, 18, ... the consistent difference between each term is 5, making it an arithmetic progression. Once you determine the first term and the common difference, you can predict any future term in the sequence using the formula: \[a_n = a_1 + (n-1) imes d\]Where:

• \(a_n\) is the nth term,
• \(a_1\) is the first term of the sequence,
• \(d\) is the common difference,
• \(n\) is the term number we are calculating.

Knowing that our sequence is arithmetic, we can confidently use this formula to compute terms like the 10th or even the 50th term with ease. Arithmetic progressions are fundamental in understanding patterns and models in mathematics.