Sequences

To enter the Sequences application, highlight the app using the directional keys. Press the ok key to enter the application. This application allows you to plot both recursive and explicit sequences, view a table of their values and compute the sum of their terms.

Getting started

Plotting sequences

To plot a sequence:

1. Highlight Add sequence.
2. Confirm with the ok key.
3. Highlight which type of expression you would like to enter: explicit expression of the sequence (based on $n$), recursive first order (expression based on the previous term) or recursive second order (expression based on the two previous terms).
4. Confirm with the ok key. The editing field will appear at the bottom of the screen.
5. Enter the expression of the sequence you wish to plot. To use the variable $n$ within the expression of the sequence, press the xnt key. To use a recursive formula, use the Toolbox menu shortcuts by pressing the toolbox key. In this case, you must indicate the value of the first term of the sequence.
6. Confirm with the ok key.
7. Highlight the Plot Graph button at the bottom of the screen or the Graph tab at the top of the screen.
8. Confirm with the ok key.

You are now in the Graph tab and your sequence has been plotted. You can move the cursor using the directional arrows and read the coordinates of the points at the bottom of the screen.

To view the options menu of the sequence on which the cursor is located, press the ok key. You can also highlight Calculate and press the ok key.

Displaying the table of values

The table of values for the sequence can be found in the Table tab. There are two ways to access it.

• First option: from the Sequences tab

  1. Highlight Display values at the bottom of the screen.
  2. Confirm with the ok key. The table is displayed.

• Second option: from any tab

  1. Highlight the Table tab at the top of the screen.
  2. Confirm with the ok key. The table is displayed.

Using the Sequences tab

Adding a sequence to the list

You can add up to 3 sequences to the list. To add a sequence to the list:

1. Highlight Add sequence at the bottom of the list.
2. Confirm with the ok key.

Then choose the type of sequence you wish to plot and enter its expression.

Editing the expression of a sequence

You can edit the expression of a sequence on the list by highlighting it and pressing the ok key.

The sequence editing field is then displayed at the bottom of the screen.

Activating or deactivating a sequence on the list

You can deactivate a sequence without completely deleting it from the sequences list. To do this:

1. Highlight the name of the sequence in the list.
2. Press the ok key to open the options menu for the sequence.
3. Highlight Show in Graph and Table and press the ok key to toggle the status of the sequence.
4. Return to the sequence list by pressing back. Repeat the steps to re-activate the sequence.

Inactive sequences will appear on the list in gray. You can edit their expressions, but these sequences will not be shown in the Graph and Table tabs.

Deleting the expression of a sequence

You can delete the expression of a sequence and enter a new one. To do this:

1. Highlight the expression of the sequence you want to delete.
2. Press the del key. The expression has been deleted.

Removing a sequence from the list

You can permanently remove a sequence from the list. To delete a sequence:

1. Highlight the name of the sequence you wish to remove from the sequence list.
2. Press the ok key to open the options menu for the sequence.
3. Highlight Delete sequence.
4. Confirm with the ok key. The sequence will disappear from the list.

You can also delete a sequence from the list by highlighting the name of a sequence and pressing the del key.

Changing the type of sequence

There are three types of expressions to use for your sequences. The types of expression available for use:

• Explicit expression of the sequence: based on $n$.
• Recursive first order: expression based on the previous term.
• Recursive second order: expression based on the two previous terms.

To change the type of sequence used:

1. Highlight the name of the sequence you wish to change.
2. Press the ok key to open the options menu for the sequence.
3. Highlight Sequence type.
4. Confirm with the ok key.
5. Highlight the type of expression you would like.
6. Confirm with the ok key. The type of the sequence has been changed. If you chose to define the sequence with a recursive expression, you must provide the first term.

When you change the type of a sequence, the expression previously entered for that sequence is deleted.

Entering the expression of a recursive sequence

You can enter the expression of a recursive sequence. For example, enter: $u_{n+2}=u_{n+1}+u_{n}$. You can use the two previous terms in the expression ($u_{n+1}$ and $u_{n}$).

To enter the expression of a recursive sequence:

1. Press the toolbox key while editing your expression.
2. Highlight the term you need to complete the expression.
3. Confirm with the ok key.

You can also manually enter the term you would like: type u(n+1) for $u_{n+1}$ and u(n) for $u_{n}$.

When entering a recursive sequence, you must define the first terms of the sequence. Enter the required terms in the list below the expression of the recursive sequence.

Changing the first term index

You can change the first term index of your sequence. To do this:

1. Highlight the name of the sequence to edit.
2. Press the ok key to open the options menu for the sequence.
3. Highlight First term index.
4. Enter the desired value.
5. Confirm with the ok key.
6. Press the back key to return to the Sequences tab. Your first term index has been changed.

Changing the color of a sequence

When you add a new sequence, a color is automatically assigned and will be used when plotting the sequence. To change the color:

1. Highlight the name of the sequence whose color you wish to change.
2. Confirm by pressing the ok key to open the Sequence options menu.
3. Choose Color and select the color of your choice.

Using the Graph tab

Moving the cursor in the graph window

You can move the cursor using the four directional arrows:

• left / right: Move the cursor to the right or left.
• up / down: Move the cursor to a sequence above or below the sequence you are on.

The cursor indicates which sequence is selected. Be sure to have the correct sequence selected before performing calculations.

Adjusting the display window

You can adjust the display window in the Graph tab to fit your needs. This includes: editing your X and Y values, zooming in and out, and more. There are three menus that allow you to adjust your graphing window: Auto, Axes and Navigate. To adjust your graphing window:

1. Highlight your desired menu. The menus are located underneath the tabs near the top of the screen.
2. Confirm by pressing the ok key.

Auto

The NumWorks graphing calculator will adapt to display the best possible view of your sequence. Once you enter your sequence, the resulting axes will be chosen automatically. When Auto is active, a yellow circle will appear to the right of its name. Once you exit this window, either with the directional keys or by changing the settings, Auto will be disabled and the yellow circle will disappear.

You can also highlight Auto and press the ok key to disabled the Auto mode.

Axes

The Axes menu allows you to make manual changes to the axes of your graphing window. To make changes to the X or Y values:

1. Highlight Axes.
2. Press the ok key to enter the menu.
3. Highlight Values of X or Values of Y
4. Confirm with the ok key.
5. Enter your desired Minimum and Maximum values. To allow the calculator to choose automatically, select Auto at the top of the screen.
6. Highlight Confirm at the bottom of the screen.
7. Confirm with the ok key.

You can make your axes equal inside the Axes menu. To do this, highlight Make axes equal and confirm with the ok key. Selecting this option will lengthen the shortest axis to match the longest. When your axes are not equal, a symbol will appear beside the Axes menu.

Navigate

Select Navigate to work with the interactive display window:

• left / up / right / down: move the window
• plus / minus: zoom in/zoom out

You can press plus and minus at anytime while in the Graph tab to zoom in/out.

Moving the cursor to a given index

1. When the cursor is on the sequence for which you are looking for a particular value, press ok.
2. The plot options menu opens, select Go to and confirm.
3. Enter the index of the point on which you want to move the cursor.
4. Select the Confirm button and press ok.

The cursor is now on the point you requested.

You can also type directly on the keyboard the value of the index to bring it to the requested point.

Calculating the sum of the terms

To calculate the sum of terms for any sequence:

1. Make sure your cursor is located on the sequence you wish to work with.
2. Press the ok key. The plot options menu will open.
3. Highlight Sum of terms.
4. Confirm with the ok key.
5. Enter the first term. Use the leftright keys to position the cursor on the first term. You can also manually enter the value of $n$ with the keyboard.
6. Confirm with the ok key.
7. Enter the last term in the same way.
8. Confirm with the ok key. You can return to the previous step by pressing the back key. If you do this, you will be asked to enter the first term again.
9. The value of the sum you would like to calculate is shown in the display bar at the bottom of the screen.

To exit the Sum of terms mode, press the ok key.

Plotting a cobweb graph

It is possible to represent a recursive sequence with a cobweb graph to study its convergence. To plot a coweb graph:

1. Make sure your cursor is located on the recursive sequence you wish to work with.
2. Press the ok key. The plot options menu will open.
3. Highlight Cobweb graph.
4. Confirm with the ok key.
5. Use the rightleft keys to navigate to the next and previous terms respectively.
6. Press the back key to exit the cobweb graph.

The plusminus keys allow you to zoom in and out around the selected term.

Using the Table tab

Modifying the interval of the table of values

When in the Table tab, you can fill the table of values with values of $n$ in any interval of your choice. There are two ways to do this:

• First option

  1. Highlight Set the interval.
  2. Confirm with the ok key. The options menu will open.
  3. Enter the values of N start and N end with the numeric keys.
  4. Enter the step value.
  5. Highlight Confirm.
  6. Confirm with the ok key. The new table will reflect the interval you have specified.

• Second option

  1. Highlight the n cell at the top of the first column.
  2. Confirm with the ok key.
  3. Highlight Set the interval.
  4. Confirm with the ok key. The options menu will open.
  5. Enter the values of N start and N end with the numeric keys.
  6. Enter the step value.
  7. Highlight Confirm.
  8. Confirm with the ok key. The new table will reflect the interval you have specified.

Entering values of n into the table

To manually enter values of $n$ into the table:

1. Highlight your desired cell in the first column (Column n).
2. Enter a value of $n$ using the numerical keys.
3. Confirm with the ok key.

Displaying the sum of terms column

You can display a column that shows the sum of the terms in the table. To view this column:

1. Highlight the cell that contains the name of your desired sequence.
2. Press the ok key. An options menu will appear.
3. Highlight Show the sum of terms.
4. Press the ok key to toggle on this feature.
5. Press the back key to return to the table. The sum of terms column will appear to the right of the sequence you selected.

If you would no longer like to view the sum of terms column, highlight the first cell of the column and press the ok key. Highlight Hide the sum of terms and press the ok key. The column is no longer displayed.

Deleting all table values

1. Highlight the n cell at the top of the first column.
2. Confirm with the ok key. An options menu will appear.
3. Highlight Clear column.
4. Confirm with the ok key. An empty value table will appear on the screen. You can now enter values in the n column manually or fill automatically.

You can delete a row from the table by highlighting it and pressing the del key.