Problem 67 - Precalculus with Limits

Question

In Exercises 67-70, write the first five terms of the sequence defined recursively. Use the pattern to write the \(n\)th term of the sequence as a function of \( n \). (Assume that \( n \) begins with 1.) \( a_1 = 6, a_{k + 1} = a_k + 2 \)

Step-by-Step Solution

Answer

The first five terms of the sequence are 6, 8, 10, 12, and 14, and the \(n\)th term of the sequence is given by \( a_n = 4 + 2n \).

Step 1: Determine the first five terms of the sequence

Starting from the first term, \(a_1 = 6\), according to the given recursive rule (\(a_{k + 1} = a_k + 2\)), the next terms can be calculated by adding 2 to the previous term. Therefore, \(a_2 = a_1 + 2 = 6 + 2 = 8\), \(a_3 = a_2 + 2 = 8 + 2 = 10\), \(a_4 = a_3 + 2 = 10 + 2 = 12\), and \(a_5 = a_4 + 2 = 12 + 2 = 14\). So the first five terms are 6, 8, 10, 12, and 14.

Step 2: Write the nth term of the sequence as a function of \( n \)

From the pattern, we notice that for each increase in \( n \), the term increases by 2. Therefore, the sequence is an arithmetic sequence with the first term \( a = 6 \) and the common difference \( d = 2 \). By the formula for the nth term \( a_n \) of an arithmetic sequence, \( a_n = a + (n - 1) * d \), we substitute \( a = 6 \) and \( d = 2 \) into the formula and we get \( a_n = 6 + (n - 1) * 2 \) or \( a_n = 4 + 2n \).

Key Concepts

Recursive Sequences

A recursive sequence is a sequence of numbers where each term is defined using the preceding terms according to a fixed rule. In the problem you encountered, the sequence is defined recursively as follows: start with an initial term, and then each subsequent term is found by applying a specific rule to the preceding term.

In mathematical terms, the given recursive sequence starts with the first term, known as the base case or initial condition:

• Initial term: \( a_1 = 6 \)

Using the recursive formula provided, \( a_{k + 1} = a_k + 2 \), you generate the next terms by adding 2 to the previous term. This results in the following computations:

• \( a_2 = a_1 + 2 = 8 \)
• \( a_3 = a_2 + 2 = 10 \)
• \( a_4 = a_3 + 2 = 12 \)
• \( a_5 = a_4 + 2 = 14 \)

The benefit of a recursive sequence is that you only need to know the rule and the initial term to find the sequence, making it straightforward to compute each new term from its predecessor.

General Term of a Sequence

The general term of a sequence provides a way to find the value of the sequence at any position without needing to calculate all preceding terms. This is particularly helpful when you need to find a term located far into the sequence.

For arithmetic sequences like the one in this exercise, the general term can be determined by observing the pattern of changes from one term to the next. From your work above, the sequence increases by a constant value, and the pattern follows a clear arithmetic progression. A general term for an arithmetic sequence can be found using the formula:

• \( a_n = a + (n - 1) \, d \)

Where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Applying this to your sequence:

• First term: \( a = 6 \)
• Common difference: \( d = 2 \)

Thus, the general term is:

• \( a_n = 6 + (n - 1) * 2 \)
• Which simplifies to: \( a_n = 4 + 2n \)

This formula allows you to directly compute the value of any term \( a_n \) in the sequence just by inserting the appropriate \( n \).

Common Difference in Sequences

The common difference is a key characteristic of an arithmetic sequence. It represents the amount you add to each term to get the next term. This constant difference between consecutive terms is what makes an arithmetic sequence linear and easy to work with.

In your sequence, the common difference \( d \) was specified in the recursive formula, "\( a_{k + 1} = a_k + 2 \)". Here, the common difference is clearly 2.

Recognizing the common difference is crucial for identifying arithmetic sequences, constructing recursive formulas, and creating general term formulas. In practical terms, the common difference can be:

• The step size that increments each term in the sequence.
• Used to express the increase or decrease per term for economic or financial calculations.

Understanding the common difference helps to pinpoint the nature of the sequence and streamline the calculation of terms, offering a swift route to unraveling more extensive sequence computations efficiently.