Question

Determine the \(n^{\text {th }}\) term of the given sequence. $$4,7,10,13,16, \ldots$$

Short answer

The \(n^{\text{th}}\) term is \(3n + 1\).

Step-by-step solution

Identify the Pattern

The given sequence is 4, 7, 10, 13, 16, ..., which is an arithmetic sequence. To identify the pattern, calculate the difference between consecutive terms: \( 7 - 4 = 3 \), \( 10 - 7 = 3 \), \( 13 - 10 = 3 \), and \( 16 - 13 = 3 \). Therefore, the common difference \(d\) is 3.

Use the Arithmetic Sequence Formula

The formula for the \(n^{\text{th}}\) term of an arithmetic sequence is \( a_n = a_1 + (n-1) d \), where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = 4\) and \(d = 3\).

Substitute the Known Values

Substitute \(a_1 = 4\) and \(d = 3\) into the formula: \[ a_n = 4 + (n-1) \times 3 \].

Simplify the Expression

Simplify the expression \(a_n = 4 + (n-1) \times 3\): \[ a_n = 4 + 3n - 3 = 3n + 1 \].

Conclusion

The general term (\(n^{\text{th}}\) term) of the sequence is \(a_n = 3n + 1\).

Key concepts

Common Difference

An arithmetic sequence contains numbers that follow a regular pattern. This pattern is determined by something called the "common difference." The common difference is simply the amount between each pair of consecutive numbers in the sequence. For the sequence given in the exercise: 4, 7, 10, 13, 16, ..., the common difference is 3.

• To identify it, subtract the first term from the second: 7 - 4 = 3.
• Continue this subtraction with other consecutive terms like: 10 - 7 = 3, and so on.

Since the difference is consistent between every two terms, we know that the common difference is indeed 3. This tells us that the sequence numbers are increasing evenly by 3 each step. This step helps us understand the structure of the sequence, which is crucial for finding any term within it.

Nth Term Formula

To find any term in an arithmetic sequence, we use a handy tool called the "Nth term formula." This formula helps predict any number in the sequence by using a simple calculation.The Nth term formula is: \[ a_n = a_1 + (n - 1) \times d \] where:

• \(a_n\) represents the Nth term we want to find,
• \(a_1\) is the first term in the sequence (4 in our case), and
• \(d\) is the common difference (3 in this sequence).

This formula uses the first term and the common difference to determine the position of any term in the sequence. Simply plug in the values you know into the formula to find the term you're looking for.

Pattern Recognition

Understanding pattern recognition is the key to identifying and working with sequences. It is like discovering the secret rule that the sequence follows. In our sequence, recognizing that each term increases by 3 reveals the arithmetic nature of the sequence. This recognition is essential because:

• It simplifies finding any term in the sequence using the Nth term formula.
• It helps to predict future terms without listing all previous numbers.

Spotting such patterns requires looking at the differences between terms and calculating them. Once identified, it allows further analysis such as finding specific terms, summing the sequence, and more.

Sequence Analysis

Sequence analysis involves breaking down the sequence to understand its properties and behavior. For arithmetic sequences, this analysis focuses on verifying the common difference and using the Nth term formula. This process helps determine key characteristics like:

• The common difference, meaning the amount between each term.
• The formula for calculating any specific term.
• Potential applications, such as predictions of future terms.

Analysis involves examining the structure and elements of the sequence, confirming calculations, and ensuring all steps fit logically. With the sequence 4, 7, 10, 13, 16,..., sequence analysis solidifies understandings like the pattern of increase and shows how to apply formulas accurately.