Problem 22 - Thomas Calculus

Question

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(2,6,10,14,18, \dots\)

Step-by-Step Solution

Answer

The formula is \(a_n = 4n - 2\).

Step 1: Identify the Pattern

Look at the given sequence: \(2, 6, 10, 14, 18, \dots\). Observe how the numbers are changing from one term to the next. Notice that each term increases by 4 from the previous one.

Step 2: Confirm the Arithmetic Sequence

Since each term is increasing by the same amount (4), this sequence is an arithmetic sequence with a common difference \(d = 4\). The first term \(a_1\) of the sequence is 2.

Step 3: Use the Formula for the n-th Term

The formula for the \(n\)-th term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \times d\).

Step 4: Plug in Values into the Formula

Substitute the known values into the formula: \(a_1 = 2\) and \(d = 4\). So, the formula becomes \(a_n = 2 + (n-1) \times 4\).

Step 5: Simplify the Expression

Simplify the expression obtained in the previous step: \(a_n = 2 + 4(n-1)\). Expand and simplify: \(a_n = 2 + 4n - 4\), which simplifies to \(a_n = 4n - 2\).

Step 6: State the Formula

The formula for the \(n\)-th term of the sequence is \(a_n = 4n - 2\).

Key Concepts

Common Difference

The common difference is a key concept in identifying and working with arithmetic sequences. It represents the constant difference between consecutive terms in the sequence. For example, consider the sequence from the original exercise: \(2, 6, 10, 14, 18, \dots\). Notice that each term increases by \(4\) from the previous one.

• The second term (\(6\)) is \(2 + 4\).
• The third term (\(10\)) is \(6 + 4\).
• The pattern continues similarly for further terms.

By consistently adding this common difference, you can generate subsequent terms in the series. Identifying the common difference \(d = 4\) confirms that this is an arithmetic sequence, making it easy to predict future numbers in the sequence without direct calculation.Understanding the common difference is crucial because it helps us not only to identify arithmetic sequences but also to derive formulas for specific terms, making complex sequences manageable with simple rules.

Formula for n-th Term

Calculating any particular term within an arithmetic sequence relies on the formula for the n-th term. Once the common difference \(d\) is determined, you can use this formula: \[a_n = a_1 + (n-1) \times d\] Here, \(a_1\) is the first term of the sequence, and \(n\) is the position of the term you're trying to find. In the step-by-step solution from the exercise, we identify the first term \(a_1\) as \(2\) and the common difference \(d\) as \(4\).

• Substituting these values in, \(a_n = 2 + (n-1) \times 4\).
• This simplifies to the formula \(a_n = 4n - 2\).

This formula allows us to plug in any position \(n\) to find the term's value in the sequence.
For instance:

• If you wish to find the 5th term, substitute \(n = 5\).
• You get \(a_5 = 4 \times 5 - 2 = 18\).

Understanding and mastering this formula provides a straightforward method for dissecting any arithmetic sequence and determining the values of its terms efficiently.

Sequence Pattern Identification

Recognizing patterns in sequences is a fundamental skill in algebra and helps to simplify problem-solving processes. By identifying these patterns, you can classify the type of sequence you are dealing with. When you encounter a sequence, examine how the terms progress from one to the next. In the exercise sequence \(2, 6, 10, 14, 18, \dots\), you observe that:

• The increment from one number to the next is consistently \(4\).
• This signifies a regular pattern, indicating it is an arithmetic sequence.

Being adept at identifying patterns entails more than just recognizing the common difference. It involves:

• Determining if the changes between consecutive terms are consistent.
• Recognizing cyclical patterns if they exist.
• Understanding if the pattern applies for all the terms or if it changes at any point.

Pattern recognition also aids in verifying your calculations and ensuring the accuracy of derived formulas, setting a foundation on which to build your understanding of sequences in mathematics.