Problem 1 - Basic Engineering Mathematics

Question

Determine the next two terms in the series: \(3,6,9,12, \ldots\)

Step-by-Step Solution

Answer

The next two terms are 15 and 18.

Step 1: Identify the Pattern

Observe the sequence given: \(3, 6, 9, 12, \ldots\). Each term increases by a constant amount. Calculate the difference between successive terms: \(6 - 3 = 3\), \(9 - 6 = 3\), and \(12 - 9 = 3\). The pattern shows that each term is obtained by adding 3 to the previous term.

Step 2: Calculate the Next Term

The last term provided in the series is 12. To find the next term, add the constant difference of 3 to this term. Thus, the next term is \(12 + 3 = 15\).

Step 3: Calculate the Second Next Term

Following the same pattern, add the constant difference of 3 to the term 15 we just calculated. Hence, the second next term is \(15 + 3 = 18\).

Step 4: Verify the Pattern

Check that the pattern holds for the terms calculated: The updated series \(3, 6, 9, 12, 15, 18\) follows the constant addition of 3 between successive terms: \(6, 9, 12, 15,\) and \(18\). This confirms the correctness of our calculations.

Key Concepts

Pattern Recognition in Arithmetic Sequences

Recognizing patterns is the first step in understanding arithmetic sequences. An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a fixed number, known as the common difference, to the previous term. To identify the pattern, you can compare consecutive terms to see what constant change they share. For example, in the sequence: \(3, 6, 9, 12, \ldots\), each term is 3 greater than the last.

• Observe the given numbers.
• Calculate the difference between successive terms.
• Confirm that the difference is constant throughout the sequence.

Once you recognize a pattern, you can use it to predict future terms or understand the sequence's behavior.

Understanding the Common Difference

The common difference is a vital component of any arithmetic sequence. It represents the fixed amount added to each term to obtain the next. In our sequence, the common difference is 3. To calculate the common difference:

• Subtract the first term from the second: \(6 - 3 = 3\).
• Repeat the process for other pairs of consecutive terms: \(9 - 6 = 3\), \(12 - 9 = 3\).

Knowing the common difference allows us to quickly find any term in the sequence by continuously adding this difference to the preceding term. It's a simple yet powerful tool to master when dealing with arithmetic sequences.

Series Continuation with Arithmetic Sequences

Continuing a series involves predicting future terms based on an established pattern. With an arithmetic sequence, continuation is straightforward due to the constant addition of the common difference. To continue the series:

• Start from the last known term.
• Add the common difference to find the next term. For our sequence starting with 12, the next term is \(12 + 3 = 15\).
• Repeat the process to find further terms: \(15 + 3 = 18\).

By following this method, you ensure accuracy and maintain the consistency of the arithmetic sequence across both known and new terms.