Follow these steps to evaluate a finite sequence defined by...

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI- $84,$ do the following. $\cdot$ In the home screen, press $[2 \mathrm{ND}]$ LIST. $\cdot$ Scroll over to OPS and choose "seq(" from the dropdown list. Press [ENTER]. $\cdot$ In the line headed "Expr:" type in the explicit formula, using the $[\mathrm{X}, \mathrm{T}, \theta, n]$ button for $n$ $\cdot$ In the line headed "Variable:" type in the variable used on the previous step. $\cdot$ In the line headed "start:" key in the value of $n$ that begins the sequence. $\bullet$ In the line headed "end:" key in the value of $n$ that ends the sequence. $\cdot$ 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. $\cdot$ In the home screen, press $[2 \mathrm{ND}]$ LIST. $\cdot$ Scroll over to OPS and choose "seq(" from the dropdown list. Press [ENTER]. $\cdot$ Enter the items in the order "Expr", "variable", "start", "end" separated by commas. See the instructions above for the description of each item. $\cdot$ Press IENTER] to see the listof tems for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. List the first four terms of the sequence $a_{n}=5.7^{n}+0.275(n-1) !$

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI- $84,$ do the following.
$\cdot$ In the home screen, press $[2 \mathrm{ND}]$ LIST.
$\cdot$ Scroll over to OPS and choose "seq(" from the dropdown list. Press [ENTER].
$\cdot$ In the line headed "Expr:" type in the explicit formula, using the $[\mathrm{X}, \mathrm{T}, \theta, n]$ button for $n$
$\cdot$ In the line headed "Variable:" type in the variable used on the previous step.
$\cdot$ In the line headed "start:" key in the value of $n$ that begins the sequence.
$\bullet$ In the line headed "end:" key in the value of $n$ that ends the sequence.
$\cdot$ 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.
$\cdot$ In the home screen, press $[2 \mathrm{ND}]$ LIST.
$\cdot$ Scroll over to OPS and choose "seq(" from the dropdown list. Press [ENTER].
$\cdot$ Enter the items in the order "Expr", "variable", "start", "end" separated by commas. See the instructions above for the description of each item.
$\cdot$ Press IENTER] to see the listof tems for the finite sequence defined. Use the right arrow key to scroll through the list of terms.
For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first four terms of the sequence $a_{n}=5.7^{n}+0.275(n-1) !$

Key Concepts

Factorial Function

The factorial function, denoted by an exclamation mark (!), is a mathematical operation that multiplies a positive integer by all the positive integers less than it. Factorials are commonly encountered in permutations, combinations, and other areas of discrete mathematics, and they play a role in modifying sequence formulas and their growth rates.

Numerical Approximation and Rounding

Numerical approximation is the process of finding approximate values for mathematical expressions, especially when an exact answer is impractical. Rounding, such as rounding to the nearest thousandth, is a common method used to simplify these approximations while retaining a desired degree of accuracy, which is essential in both theoretical analysis and digital computations.

Calculator Sequence Functions

Modern calculators, such as those from the TI series, often have built-in functions to generate sequences. These functions allow users to input an expression, specify the variable, and define the start and end values for the sequence index. This capability assists in visualizing and analyzing the behavior of sequences quickly and accurately.

Finite Sequences

A finite sequence is an ordered list of numbers where the number of elements is limited to a specific count. They are defined by listing each term or by using a formula that computes the nth term based on its position in the sequence. Understanding finite sequences is fundamental in both discrete mathematics and numerical analysis.

Explicit Sequence Formula

An explicit formula for a sequence provides a direct way to compute any term in the sequence as a function of its position. Unlike recursive formulas, explicit formulas allow for the calculation of the nth term without needing to compute preceding terms, making them efficient for understanding and analyzing sequences.

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