Problem 58 - 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 \(15^{\text { th }}\) term of the sequence \(a_{1}=625, a_{n}=0.8 a_{n-1}+18.\)

Step-by-Step Solution

Answer

The 15th term is approximately 101.32.

Step 1: Input Initial Term

On the graphing calculator, start by inputting the initial term of the sequence. Enter \(a_1 = 625\) and press the [ENTER] key to store this as the first term.

Step 2: Enter the Recursive Formula

Input the recursive formula into the calculator. Begin by pressing [2ND] and then inputting ANS to represent the previous term \(a_{n-1}\) in the formula. Key in the expression \(0.8 \times \text{ANS} + 18\) and press [ENTER] to store the recursive relationship.

Step 3: Compute Successive Terms

Continuously press [ENTER] to compute each successive term using the recursive formula. After each press, the calculator will use the last computed value and apply the recursive formula to find the next term.

Step 4: Find the 15th Term

Press [ENTER] repeatedly until you reach the 15th computation. This result is the 15th term of the sequence. Based on the recursive evaluation, the 15th term \(a_{15}\) is 101.32494464000003.

Key Concepts

Graphing Calculator

A graphing calculator is an invaluable tool for solving sequences, especially when dealing with recursive sequences. This type of calculator is capable of storing a sequence and allows for iterative calculations right on its interface.
To effectively use a graphing calculator for evaluating recursive sequences:

• Access the home screen for easy input and management of terms and formulas.
• Use the [2ND] ANS function to call upon previous terms within the sequence, saving time on manual computation.
• Repeatedly pressing [ENTER] automates the calculation, reducing the risk of manual errors.

This step-by-step system simplifies complex sequences, making evaluation straightforward and consistent, especially in lengthy or multifaceted calculations.

Initial Term

The initial term in a recursive sequence, often denoted as \(a_1\), is the starting point from which all other terms are derived. It's crucial because it sets the foundation for the sequence. Without a clearly defined initial term, the sequence cannot be properly evaluated.
To find the initial term in the provided exercise, you would input \(a_1 = 625\) into the calculator. This establishes the baseline value. Every subsequent calculation of terms in the sequence is built from this initial step. Correctly inputting \(a_1\) ensures that the entire sequence develops as specified by the recursive formula.

Recursive Formula

A recursive formula defines each term in a sequence based on the preceding term or terms. It is a systematic way of developing terms where each new term is constructed by applying the formula to the previous terms.
For example, in the given exercise, the formula is \(a_n = 0.8a_{n-1} + 18\). This indicates that each term is 80% of the previous term plus 18. The recursive formula is crucial to understanding the direction and growth of the sequence, as it dictates how each term is formed.
When entered correctly in a graphing calculator, it uses the ANS key to dynamically reference the latest term, ensuring consistency throughout the sequence evaluation process.

Sequence Evaluation

Sequence evaluation involves iteratively applying the recursive formula to calculate successive terms. This is done using the graphing calculator for automation and accuracy.
To evaluate a sequence, follow these steps:

• Begin with entering the initial term.
• Input and store the recursive formula.
• Start evaluating by pressing [ENTER] repeatedly.

Through these steps, you can find any term in the sequence, such as the 15th term as in the original exercise. Each press of [ENTER] applies the recursive formula to the latest known term, calculating the next one. By this method, the 15th term of the sequence, calculated as 101.32494464, illustrates the gradual transformation through each step of the recursive process.