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Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following. 鈥� In the home screen, press [2ND] LIST. 鈥�Scroll over to OPS and choose "seq(" from the drop down list. Press [ENTER]. 鈥� In the line headed 鈥淓xpr:鈥� type in the explicit formula, using the \([\mathbf{X}, \mathbf{T}, \boldsymbol{\theta}, \boldsymbol{n}]\) button for n \(\cdot\) In the line headed "Variable:" type in the variable used on the previous step. 鈥� In the line headed 鈥渟tart:鈥� key in the value of n that begins the sequence. 鈥� In the line headed 鈥渆nd:鈥� key in the value of n that ends the sequence. 鈥� Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. 鈥� In the home screen, press [2ND] LIST. 鈥� Scroll over to OPS and choose 鈥渟eq(鈥� from the drop down list. Press [ENTER]. 鈥� Enter the items in the order 鈥淓xpr鈥�, 鈥淰ariable鈥�, 鈥渟tart鈥�, 鈥渆nd鈥� separated by commas. See the instructions above for the description of each item. 鈥� Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. List the first six terms of the sequence \(a_{n}=\frac{n !}{n}\)

Step by step solution

01

Set Up Expression on TI Calculator

Navigate to the home screen of your TI-84 or TI-83 calculator, then press [2ND] and select LIST. Scroll to OPS and choose 'seq( '. When prompted for 'Expr:', input the formula `n!/n`, making sure to use the [x], [T], [胃], [n] button to represent `n`. Use `n` specifically when entering the expression.

02

Enter Variable Name

For the line 'Variable:', confirm that you have entered 'n' since that is the variable used in your sequence formula.

03

Set the Sequence Range

Input the starting and ending values for 'n'. In this problem, the sequence begins at n=1 and ends at n=6, so enter '1' for 'start' and '6' for 'end'.

04

Generate and View Sequence

On your TI calculator, press [ENTER] multiple times until you see the sequence syntax on your home screen. After pressing [ENTER] one more time, the list of terms in your sequence will appear. Use the right arrow key to scroll through these terms.

05

Calculate Each Sequence Term Manually

If doing manually or confirming, compute each term using the formula. For instance, for n=1, calculate: \[ a_1 = \frac{1!}{1} = 1 \]For n=2, calculate:\[ a_2 = \frac{2!}{2} = 1 \]Continue similarly for n=3, n=4, n=5, n=6.

06

Round and List Terms

Rounding may be required, but in this scenario, none of these calculations need rounding. List the terms as calculated: 1, 1, 2, 6, 24, 120.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Formula

A sequence formula is a mathematical expression that defines the terms of a sequence. In the context of this exercise, the explicit formula given is \(a_{n} = \frac{n!}{n} \). This formula denotes that each term in the sequence can be calculated by taking the factorial of the term number \(n\) and dividing it by \(n\) itself. The explicit formulation gives us a straightforward way to directly calculate any term from the sequence without needing the preceding terms. This is particularly useful for both manual calculations and programming sequences in calculators like the TI-83 and TI-84, because it simplifies the determination of any finite sequence term.

TI-84 Calculator Instructions

The TI-84 calculator provides a handy set of steps to evaluate finite sequences using an explicit formula. To process sequences:

• Access the home screen, then press [2ND] and select LIST.
• Navigate to OPS and choose 'seq('.
• Input the formula for the sequence, using the [x], [T], [胃], [n] button to denote the variable.
• Enter 'n' as the variable name, then specify the starting and ending values (for this example, start at 1 and end at 6).
• Confirm your entries by pressing [ENTER] until you see the sequence syntax, and press [ENTER] again to generate the terms.

This method allows you to quickly evaluate the entire sequence and view all terms through the calculator display.

TI-83 Calculator Instructions

Using a TI-83 calculator to find the terms of a sequence follows similar steps but with a slightly different interface. Here's how to evaluate a sequence:

• Begin at the home screen, press [2ND] and select LIST.
• In OPS, choose 'seq('.
• Enter the formula using the sequence format: 'Expr', 'Variable', 'start', 'end'. Separate each with commas.
• Press [ENTER] to view the calculated sequence terms.

This allows for a straightforward, step-by-step determination of sequence terms using the capabilities of the TI-83 calculator, providing you the sequence conveniently on-screen.

Factorial Calculation

A factorial is a product of an integer and all the integers below it, denoted as \(n!\). For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). Factorials grow rapidly with larger integers, making them useful in many mathematical sequences and permutations. In this exercise, the calculation of \(n!\) is crucial for computing each term using the formula \(a_{n} = \frac{n!}{n}\). This means for each positive integer \(n\), the factorial has to be computed first. The resulting value is then divided by \(n\) to determine the sequence term, leveraging the factorial's rapid growth to illustrate significant differences between consecutive terms.

Explicit Formula

An explicit formula allows you to determine the \(n^{th}\) term of a sequence directly. Unlike recursive formulas, where each term is based on the previous term, an explicit formula provides immediate answers without requiring the computation of preceding terms. For example, given \(a_{n} = \frac{n!}{n}\), you can plug any value of \(n\) into this formula to get the sequence term instantly. This direct relationship is efficient for evaluating terms quickly, especially with finite sequences where knowing non-consecutive terms might be necessary. This formula type simplifies pattern recognition and computation.

Mathematics Rounding

Rounding is essential when dealing with non-integer results, especially in sequences. While this exercise had integer outcomes, understanding the principles of rounding can aid in cases where precision is key. Rounding to the nearest thousandth involves looking at the digit in the fourth decimal place and determining whether to round up or remain the same based on standard rounding rules:

• If the digit is 5 or more, round up the third decimal place by one.
• If the digit is less than 5, leave the third position unchanged.

Effective rounding ensures accuracy in various applications, maintaining consistency when terms need to conform to specific decimal precisions.