Problem 57 - 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 five terms of the sequence \(a_{1}=\frac{87}{111},\) \(a_{n}=\frac{4}{3} a_{n-1}+\frac{12}{37} .\) Use the \(>\) Frac feature to give fractional results.

Step-by-Step Solution

Answer

First five terms are \( \frac{87}{111}, \frac{446}{333}, \frac{1694}{999}, \frac{5954}{2997}, \frac{20716}{8991} \) as fractions.

Step 1: Enter Initial Term

On your graphing calculator, go to the home screen. Key in the initial term, which is given as \( a_1 = \frac{87}{111} \). Press the [ENTER] key to store this value as the starting point of the sequence.

Step 2: Enter the Recursive Formula

Now, input the recursive formula. Begin by keying the fraction \( \frac{4}{3} \) and press [MULTIPLY]. Then, press [2ND] followed by [ANS] to refer to the previous result, \( a_{n-1} \). Finally, add the fraction \( \frac{12}{37} \). The input should thus look like \( \frac{4}{3} \times \mathtt{ANS} + \frac{12}{37} \). Press [ENTER] to store this formula.

Step 3: Calculate Second Term

Press [ENTER] once more. The value displayed will be the second term of the sequence, \( a_2 \). Make sure to convert the result into a fractional form using the > Frac feature if it's not already in a fraction.

Step 4: Calculate Third Term

Press [ENTER] again to compute \( a_3 \), repeating the process. Again, use the > Frac feature as needed to ensure the result is expressed as a fraction.

Step 5: Calculate Fourth Term

Repeat by pressing [ENTER] to compute \( a_4 \). Convert into a fraction if required using the > Frac function.

Step 6: Calculate Fifth Term

Press [ENTER] once more to calculate \( a_5 \). Use the > Frac feature to convert the result into a fraction if it's not already displayed in that form.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for solving recursive sequences. It allows for quick computations and can handle complex calculations beyond basic arithmetic. By inputting the initial term and recursive formula, the calculator automates the process of finding successive terms in a sequence.
Using a graphing calculator, you can efficiently evaluate terms in a sequence by simply pressing the "ENTER" key after setting up the recursive formula. This saves time and reduces the chance of error when compared to manual calculations. Additionally, many graphing calculators come with features to convert results into fractional form, which can be particularly handy when dealing with precise or non-integral outputs.

• Start by ensuring your calculator is in the home screen mode.
• Input the initial value and store your recursive function.
• Press "ENTER" repeatedly to generate each new term.

Initial Term

The initial term in a recursive sequence is the starting point from which all other terms are derived. It acts as the foundation, and any subsequent computation uses this value as the baseline. For our specific example, the initial term is given as \( a_1 = \frac{87}{111} \).
To input the initial term on a graphing calculator, simply type in the fraction or number provided and press "ENTER". This value is now stored in your calculator as the first term. Without this starting value, it's impossible to continue calculating further terms. Remember, the initial term is crucial to obtain the correct sequence since every term depends on it.
Depending on the problem, the initial term might come as a fraction, decimal, or whole number. Always check whether the exercise requires input as a fraction to ensure accuracy when calculating later terms.

Recursive Formula

A recursive formula defines each term in the sequence using one or more previous terms. It is a guideline that lets you calculate the next value based on current known values. The recursive formula in our example is \( a_{n} = \frac{4}{3} a_{n-1} + \frac{12}{37} \).
A recursive formula will often involve operations such as multiplication, division, and addition with the previous term. In this case, the expression multiplies the previous term \( a_{n-1} \) by \( \frac{4}{3} \) and adds \( \frac{12}{37} \).

• When entering the recursive formula on a calculator, use the "ANS" key to refer to \( a_{n-1} \), the previous result.
• Ensure the entire expression is correct before pressing "ENTER" to store and use for future calculations.
• This formula is repeated as needed to generate terms until the desired term is reached.

The recursive formula offers a systematic approach to extending a sequence step by step, illustrating how each term is derived from its predecessor.

Fractional Results

When working with recursive sequences, results can often be fractions. It's important to understand how to interpret and work with these fractions, especially when precision is crucial. In our exercise, using fractions ensures we maintain accuracy.
Graphing calculators often provide a feature to convert decimal results into fractions, usually denoted by the "> Frac" button. This conversion is particularly useful when the recursive formula or initial term involves fractions, as it preserves the exactness of the result.

• After computation, check if the result is in decimal form.
• Use the "> Frac" function to convert it back to a fractional equivalent.
• Fractional results offer a more accurate representation as opposed to rounded decimals.
• This precision is vital when each subsequent term is calculated based on previous fractional results, reducing rounding errors.

Understanding how to effectively manage fractional results allows for an accurate and reliable solution in a sequence that builds on recursive calculations.