Problem 60 - College Algebra

Question

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1}.\) Press \([\mathrm{ENTER}].\) • Continue pressing \([\text { ENTER }]\) to calculate the values for each successive term. Use the steps above to find the indicated term or terms for the sequence. Find the first ten terms of the sequence \(a_{1}=8, a_{n}=\frac{\left(a_{n-1}+1\right) !}{a_{n-1} !}.\)

Step-by-Step Solution

Answer

The first ten terms are 8, 9, 10, 11, 12, 13, 14, 15, 16, 17.

Step 1: Initial Term

Start by inputting the initial term of the sequence. The initial term for this sequence is given as \( a_1 = 8 \). Enter this value in your graphing calculator and press [ENTER].

Step 2: Enter Recursive Formula

Now, input the recursive formula \( a_n = \frac{(a_{n-1} + 1)!}{a_{n-1}!} \) using the calculator. Key in the formula by using numbers and the function keys; to use the previous term \( a_{n-1} \), key in [2ND] ANS on the calculator and then press [ENTER].

Step 3: Calculate Successive Terms

For each term from 2 to 10, press [ENTER] repeatedly to calculate each successive term based on the recursive relation. The calculator will use the value of the previous term to compute the next term.

Step 4: List Each Term

As you continue pressing [ENTER], list out each term calculated by the calculator up to the tenth term. For instance, calculate and verify that \(a_2 = 9\), \(a_3 = 10\), \(a_4 = 11\), and so on, following the recursive pattern.

Key Concepts

Initial Term

In any recursive sequence, the initial term forms the starting point. It is the first value from which all other terms are generated. In the given exercise, the initial term is provided as \( a_1 = 8 \). This means the sequence begins at 8, and it establishes the basis for subsequent calculations. To kick off the process, you input this initial term into your graphing calculator. By pressing [ENTER], the calculator acknowledges this term as the first in the sequence. This step might seem simple, but it is crucial because the initial term sets the framework for the entire recursive operation. A slight change in the initial term can drastically alter the resulting sequence.

Recursive Formula

A recursive formula is like a blueprint for constructing a sequence one step at a time. It defines how each term relates to its predecessor. For our exercise, the recursive formula is \( a_n = \frac{(a_{n-1} + 1)!}{a_{n-1}!} \). Here's how to understand it:

• \(a_{n-1}\) stands for the previous term in the sequence.
• The formula states that to find any term, you add one to the previous term, take the factorial, and divide by the factorial of the previous term.

You'll enter this formula into your graphing calculator, using each function key carefully. When the formula uses the previous term, the [2ND] ANS key on the calculator acts as a substitute for \(a_{n-1}\). This setup allows you to generate terms beyond the initial term by simply pressing [ENTER] repeatedly.

Sequence Evaluation

Evaluating a recursive sequence involves calculating successive terms based on the initial term and recursive formula. After setting the initial term, you'll use the recursive formula repeatedly. For the given sequence, it means pressing [ENTER] after inputting the recursive formula.

This process allows the calculator to compute the solutions, each of which represents a term in the sequence. As you press [ENTER], ideally, you should see terms like \(a_2 = 9\), \(a_3 = 10\), and so forth until the sequence reaches the tenth term. Each term builds upon the last, following the pattern set by the recursive formula. Remember, a clear systematic approach helps verify each result, preventing errors and ensuring all calculated terms align with the expected pattern.

Graphing Calculator Steps

Using a graphing calculator effectively allows you to visualize and compute terms in a recursive sequence with precision. Here's a handy checklist to get you started:

• First, enter the initial term of the sequence and press [ENTER].
• Next, input the recursive formula carefully, substituting \(a_{n-1}\) with [2ND] ANS.
• For each new term, just press [ENTER]. Repeat this until all desired terms are calculated.

These steps help leverage the calculator's capabilities to handle the complexities of recursive sequences. The repetitive pressing of [ENTER] might seem tedious, but it ensures each term is based squarely on its predecessor, accurately embodying the recursive nature of the sequence. Calculators eliminate cumbersome manual calculations, especially for factorials, which rapidly become intricate.