Problem 33 - Algebra 2

Question

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 5,8,11,14,17, \dots $$

Step-by-Step Solution

Answer

The next two terms in the sequence are 20 and 23. The explicit formula for the nth term is \(3n + 2\), and the recursive formula is \(a_n = a_{n-1} + 3\).

Step 1: Identify Sequence

The sequence given is 5, 8, 11, 14, 17. It is observed that each term increases by 3, indicating that it's an arithmetic sequence and the common difference \(d\) is 3.

Step 2: Find Next Two Terms

To find the next two terms, add the common difference to the last given term, 17 + 3 = 20, and then add the common difference to this new term, 20 + 3 = 23. Therefore, the next two terms are 20 and 23.

Step 3: Find Explicit Formula

The explicit formula for an arithmetic sequence is \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference. Substituting \(a = 5\) and \(d = 3\) into the formula, we get:\(a_n = 5 + (n-1)*3 = 5 + 3n - 3 = 3n + 2\). So the explicit formula for the nth term is \(3n + 2\).

Step 4: Find Recursive Formula

The recursive formula for an arithmetic sequence is \(a_n = a_{n-1} + d\). This means the nth term is found by adding the common difference to the previous term. In this case, the recursive formula is \(a_n = a_{n-1} + 3\).

Key Concepts

Common Difference

In an arithmetic sequence, each term is obtained by adding a constant value to the previous term. This constant value is known as the "common difference." Understanding the common difference is crucial because it tells us how each term in the sequence relates to the others. In our sequence, which starts with 5, 8, 11, 14, and 17, the common difference can be found by subtracting any term from the next term. For example:

• 8 - 5 = 3
• 11 - 8 = 3
• 14 - 11 = 3

This consistent increase confirms that the common difference is 3. With this information, we can predict future terms or establish formulas to express the sequence mathematically.

Explicit Formula

An explicit formula allows you to find any term in the sequence without needing to know the previous terms. It is a powerful tool because it gives you direct access to any term's value based on its position, alleviating the need for sequential calculations.

For an arithmetic sequence, the explicit formula is given by:

• \(a_n = a + (n-1)d\)

where:

• \(a\) is the first term of the sequence
• \(n\) represents the position of the term
• \(d\) is the common difference

In our example, substituting \(a = 5\) and \(d = 3\), the formula becomes:

• \(a_n = 5 + (n-1) \times 3\)
• \( = 5 + 3n - 3\)
• \( = 3n + 2\)

Thus, the explicit formula for this sequence is \(a_n = 3n + 2\), which can be used to calculate any term directly.

Recursive Formula

The recursive formula of an arithmetic sequence defines each term based on the preceding one, using the common difference. This approach is useful because it establishes a clear chain of dependency between successive terms, illustrating how the sequence progresses.

The recursive formula is expressed as:

• \(a_n = a_{n-1} + d\)

where:

• \(a_n\) is the current term
• \(a_{n-1}\) is the previous term
• \(d\) is the common difference

Using our sequence, where the first term \(a_1 = 5\) and the common difference \(d = 3\), we can write:

• \(a_n = a_{n-1} + 3\)

This formula means to find any term, you simply add 3 to the term before it, continuing the pattern established by the sequence.

Nth Term

Finding the nth term of an arithmetic sequence provides a way to identify the value of any term in the sequence based on its position. This is crucial when dealing with large sequences, where it may not be practical to list every single term.

By using the explicit formula \(a_n = 3n + 2\), derived from earlier calculations, the nth term for our sequence is simple to compute:

• for example, to find the 10th term \(a_{10}\), substitute \(n = 10\):
• \(a_{10} = 3(10) + 2\)
• = 30 + 2
• = 32

This shows that the 10th term is 32, demonstrating how the explicit formula provides a straightforward method to reach any term directly. Regardless of the term you’re seeking, understanding how to compute the nth term lets you tap into any part of the sequence with ease.