Problem 41 - 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. $$ 21,13,5,-3, \dots $$

Step-by-Step Solution

Answer

The next two terms are -11 and -19. The explicit formula is \( a_n = 21 + (n - 1) * (-8) \), and the recursive formula is \(a_n = a_{n-1} - 8\) for \(n > 1\), \(a_1 = 21\)

Step 1: Identify the Constant Difference

By subtracting subsequent terms, we can identify the constant difference. In our case, for example subtracting 13 from 21 we get 8. If we subtract 5 from 13, we also get 8. So the constant difference in this sequence is -8.

Step 2: Predict the next Term

The next term in the series can be identified by subtracting the constant difference, -8 from the last known term. That is, -3 (last term) - 8 = -11. Repeating the step, -11 - 8 = -19. So, the next two terms are -11 and -19.

Step 3: Formulate the Explicit Formula

The explicit formula for an arithmetic sequence is \(a_n = a_1 + (n - 1) * d\) where a_n is the n-th term, a_1 is the first term, n is the position of the term and d is the common difference. Substituting the known values, we get: \(a_n = 21 + (n - 1) * (-8)\). This is the explicit formula for this sequence.

Step 4: Formulate the Recursive Formula and Identify the Formulas

The recursive formula for an arithmetic sequence of the form 'each term is obtained by adding a constant difference to the previous term' can be written as follows: \(a_n = a_{n-1} + d\) for \(n > 1\), and \(a_1 = 21\). Here, \(a_n\) is the \(n^\text{th}\) term, \(a_{n-1}\) is the previous term, and \(d\) is the common difference. Substituting the known values, we get: \(a_n = a_{n-1} - 8\) for \(n > 1\) and \(a_1 = 21\). This is the recursive formula. The explicit formula is : \(a_n = 21 + (n - 1) * (-8)\) and the recursive formula is : \(a_n = a_{n-1} - 8\) for \(n > 1\), \(a_1 = 21\).

Key Concepts

Explicit Formula

The explicit formula is a powerful tool for finding any term in an arithmetic sequence without needing to know the preceding terms. It directly calculates the value of the n-th term using a general formula. For the arithmetic sequence given: 21, 13, 5, -3, the explicit formula is derived as follows:

- Start with the first term, known as \(a_1\), which is 21 in this sequence.- Use the common difference \(d\), which is -8 (found by subtracting each term from the previous one).- The explicit formula is represented as: \[ a_n = a_1 + (n - 1) imes d \]
Substitute the known values:- \( a_n = 21 + (n - 1) \times (-8) \). - This makes it easy to find any term in the sequence by plugging in the term number \(n\). For example, to find the 10th term, replace \(n\) with 10.

This formula provides a quick way to calculate any term in the sequence efficiently.

Recursive Formula

Recursive formulas are different because they rely on the previous term to find the next one. This step-by-step approach means you start from the first term and use the formula repeatedly to get to the term you want.

For our sequence:- We started with the first term \(a_1\), which is 21.- The recursive formula is expressed as: - \( a_1 = 21 \) - \( a_n = a_{n-1} + d \) for \(n > 1\)

With our sequence's common difference \(d = -8\), the recursive formula becomes:- \( a_n = a_{n-1} - 8 \) for \( n > 1 \)
This method is particularly useful when you're building the sequence step by step and don't need the entire sequence all at once, like when making a list of numbers in order.

Common Difference

The common difference is the constant amount you add (or subtract) to one term to get to the next term in an arithmetic sequence. This sequence attribute helps define the nature and behavior of the sequence. It’s essential in forming both explicit and recursive formulas, as shown in our example.

To find the common difference: - Subtract any term from the previous term. - In the sequence 21, 13, 5, -3, subtract 13 from 21, and you get -8. - Verify by subtracting 5 from 13 and -3 from 5. Each gives -8, proving it’s consistent.
The common difference in this sequence is -8, indicating the sequence decreases steadily. Understanding the common difference is crucial in solving any arithmetic sequence problems as it precisely dictates the step-by-step progression of the terms.