Casio fx-991CW: Recurrence Sequences

Casio fx-991CW: Recurrence Sequences

Introduction

Today’s edition we will work with recurring sequences, which can take the form of:

u_n = f(u_n-1), with initial condition u0

u_n = f(u_n-1, u_n-2) with initial conditions u0, u1

In general:

u_n = f(u_n-1, u_n-2, … u_n-k) with initial conditions u0, u1, u2, …, u_n-(k-1)

There is no app on the Casio fx-991CW that is dedicated to be recurrence sequences.

Recurring Sequences the Quick and Easy Way with the Calculate App

A quick way to tackle recurrence sequences and involves the use of the last answers key ( [Ans] ). This works on recurrence sequences with only one initial condition, u0.

Start by entering u0 and press [ EXE ]. Then enter the recurrence function, using Ans as u_(n-1).

Example:

u_n = 0.75 * u_(n-1)^2 + 1.03 * u_(n-1) with u0 = 1

Settings: MathI/DecimalO (Input/Output), Number Format (Norm 1)

1 [ EXE ]

0.75×Ans²+1.03×Ans [ EXE ] (Disp: 1.78)

[ EXE ] (Disp: 4.2097)

[ EXE ] (Disp: 17.62717157)

[ EXE ] (Disp: 251.1938698)

[ EXE ] (Disp: 47582.49986)

…

Recurrence Sequences with the Spreadsheet App

A way to generate recurrence sequences is to use the Spreadsheet app. The app contains a spreadsheet of size 5 columns (A, B, C, D, E) by 45 rows. Depending on the number of the initial conditions, up to 44 terms can be generated.

For the purposes of this blog, we’ll use column A, but we can use any column, as long as there is memory in the Spreadsheet.

u_n = f(u_n-1), with initial condition u0

Set A1 as the value of u0.

At cell A2:

1. Press [ TOOLS ].

2. Select Fill Formula and press [ OK ].

3. Enter the recurrence function, using A1 as u_(n-1).

4. At the range, set as A2:A#, to create a list of #-1 terms. For example, the range A2:A11 to create a list of 10 terms.

5. Select Confirm and press [ OK ].

Example:

Under Show Cell, Value (using the [TOOLS] menu)

u_n = (u_(n-1) + 0.5) * u_(n-1) with u0 = 0.65 for 7 terms.

Cell A1 = 0.65

Cell A2:

Fill Formula:

Form = (A1+0.5)×A1

Range: A2:A8

	A (display)
1	0.65
2	0.7475
3	0.93250625
4	1.335821031
5	2.452328343
6	7.240078475
7	56.03877556
8	3168.363754

u_n = f(u_n-1, u_n-2), with initial condition u0, u1

Set A1 as the value of u0.

Set A2 as the value of u1.

At cell A3:

1. Press [ TOOLS ].

2. Select Fill Formula and press [ OK ].

3. Enter the recurrence function, using A1 as u_(n-2) and A2 as u_(n-1).

4. At the range, set as A3:A#, to create a list of #-2 terms. For example, the range A2:A11 to create a list of 9 terms.

5. Select Confirm and press [ OK ].

Example:

u_n = u_(n-1) + u_(n-2), u0 =1 , u1 = 1 for 7 terms.

Yes, this is ripe for the Fibonacci sequence.

Cell A1 = 1

Cell A2 = 1

Cell A3:

Fill Formula:

Form = A1+A2

Range: A3:A9

	A (display)
1	1
2	1
3	2
4	3
5	5
6	8
7	13
8	21
9	34

Until next time,

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Casio fx-991CW: The Spreadsheet

Casio fx-991CW: The Spreadsheet

Welcome to the Casio fx-991CW segment, which is planned to be every last Saturday of the month for 2025!

Spreadsheet

Screenshots are used the classpad.net website which contains the fx-991CW emulator.

The Casio fx-991CW has a Spreadsheet mode. The spreadsheet is 45 rows and 5 columns. The spreadsheet has a memory capacity of 2,380 bytes. The spreadsheet has basic spreadsheet functions. Data is saved as long as the calculator is in the Spreadsheet mode. Individual results can be stored in the calculator’s memory registers A, B, C, D, E, F, x, y, and z.

The spreadsheet of the fx-991CW can only contain numerical data. There are no labels.

Auto Calc is assumed to be on.

Sum, Mean, and Count of a Range of Cells

The sum of a range: = Sum( α# : β# )

The arithmetic mean of all the numbers in a range: = Mean( α# : β# )

The count of all cells that have a value stored: = Sum( α# : β# ) ÷ Mean( α# : β# )

The count has to be constructed because there is no count function in the fx-991CW. Since Sum is Σx and Mean is Σx / n. Hence count, or n, is: Count = Sum / Mean = Σx / (Σx / n) = Σx * (n / Σx) = n.

The Sum, Count, and the colon character ( : ) are found in the Catalog.

Example:

	A	B
1	45	88
2	25	76
3	93	39

Min, Max, and Statistical Range of a Range of Cells

The minimum value of a range of cells: = Min( α# : β# )

The maximum value of a range of cells: = Max( α# : β# )

The statistical range of a range of cells: = Max( α# : β# ) - Min( α# : β# )

Example:

	A	B
1	45	88
2	25	76
3	93	39

Multiply a Range of Cells with a Constant Rate

We can apply a formula to a range of cells through the Tools menu. The Fill Formula dialogue:

Fill Formula

Fill = (start the formula with the cell you are currently at)

Range : α# : β#

> Confirm

To make a cell absolute, attack a dollar sign, to both the column letter and row number. The dollar sign ( $ ) is in the Catalog.

Example:

	A	B
1		0.85
2	59.95
3	42.75
4	73.95

Storing a Cell’s Value

Exiting the Spreadsheet mode or turning the calculator off will erase the contents of the Spreadsheet. A way to store values permanently is to store the cell’s contents in one of the variables (A – F, x, y, z). To store a value:

1. Go to the cell.

2. Press the [ VARIABLE ] button.

3. Select a variable and press [ EXE ].

4. Select Store.

The value is now stored in the variable, which can be used in other modes. Unlike the spreadsheet, values stored in values are kept until a reset.

I hope you find this helpful.

Until next time,

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Review: Casio fx-991CW (and the lack of multi-statements)

Review:  Casio fx-991CW (and the lack of multi-statements)

Quick Facts

Model:  fx-991CW

Company:  Casio

Years:  Internationally: 2022 - present, United States: April 2023 - present

Type:  Scientific

Power:  Solar with 1 LR44 backup battery

Display:  Multiline, 4-color gray display

Original Price:  $21.00 - $22.99 U.S. Dollars, varies on the specific version

Number of Variables:  9

Operating System: Algebraic

Update on the Classwiz

The fx-991CW is an update in Casio's Classwiz series, specifically on the 2015 fx-991EX Classwiz.  

Check out my review on the fx-991EX Classwiz from November 2015 here:

http://edspi31415.blogspot.com/2015/11/casio-fx-991ex-classwiz-review.html

The modes on the Classwiz are:

Calculate:  the main app for mathematical calculations

Statistics:  1 to 2 variable statistics.  Regressions include:

Linear:  y=a+bx

Quadratic: y=a+bx+cx^2

Logarithmic:  y=a+b*ln(x)

Exponential:  y=a*e^(bx)

Power I:  y=a*b^x

Power II: y=a*x^b

Inverse:  y=a+b/x

Distribution:  Calculations involving the binomial, normal, and Poisson.  The Inverse Normal deal with the lower tail (left) probability only.  

Spreadsheet:  Like the fx-991EX and Casio's current graphing calculators, the fx-991CW has a spreadsheet which has a 5 x 45 cell capacity.  The total memory is increased to 2,380 bytes (from 1,700 bytes on the fx-991EX).  Basic spreadsheet features included are:  labeling cells, sum, mean, minimum, maximum, absolute cell references ($), copy, cut, and paste.  

Table:   Generate a table of one function or two functions.   The functions are defined as f(x) and g(x).  With a QR code, you can generate a graph of these functions.

Equation:   Solve linear systems, up to 4 x 4 equations.  Solve polynomials, up to 4th order.  Real coefficients only but complex roots are calculated.   The general equation solver is moved to this mode.   

Inequality:  Solve inequalities up for polynomials of orders 2, 3, and 4.

Complex Number:  Complex number arithmetic with polar/rectangular conversion, integer powers, real/imaginary parts, conjugate

Base N:  Integer arithmetic with Boolean logic.  Instead of keys, the [FORMAT] key cycles through the four modes: Decimal, Hexadecimal, Binary, Octal.   Binary integers are up to 31 bytes with 1 sign bit.  We can not store or recall varaible values in this mode.

Matrix:  Works with four matrices up to 4 rows and 4 columns.  Functions include transpose, inverse, and determinant.

Vector:  Works with 4 vectors with either 2 or 3 elements.  Functions include dot product, norm, and cross product.

Ratio:  Solves common ratio problems.

Math Box:  This is the a new feature to the Classwiz.   For the United States, the Math Box has two simulations:  Dice Rolls and Coin Toss.   Depending on the country, the CW may include additional features.  

Emulator and Classpad.net

Purchasing a fx-991CW came with a emulator license with ClassPad.net.   The license number can be obtained using the Get Started option from the Settings menu.   Use the QR code.  For my calculator, the license for using ClassPad.net is seven years at no cost.  However, using an emulator without first purchasing an eligible calculator will require an additional cost.  

You can find details about Classpad.net here:  https://classpad.net/intl/features/

Keyboard, What is Added, and What is Subtracted

The keyboard of the fx-991CW is quite different from the previous Casio calculators.   The keys are now round with a lot less labels.   Most of functions are now stored in the catalog and tools.  Let's go over some of these keys:

[ house icon ]: [HOME].   This is where we switch the modes of the fx-991CW.   There are no numerical shortcuts, so we have to arrow and scroll to select the mode we want.  

[ three lines ]: [SETTINGS].  The settings key replaces the SETUP key sequence.   

Also note that the sub menus are selected by either pressing [ → ] or [ EXE ].  If there are radio buttons, select the option desired.  

[ curved arrow ]: [EXIT/EDIT].  This key, on the 2nd row, 2nd key from the left, will be used to exit menus and re-edit expressions.   

[ double up arrow ]/[ double down arrow ]:  This key, top right of the calculator, is used to quickly scroll through menus or lists.  Think of this key as the Page Up/Page Down key.

[ < >x] ]/[VARIABLE]:  This is the variable key.

Here is where we will store and edit values that are stored in each of the nine variables.   This key replaces the STORE key.     If we are operate in a mode that does not allow for editing or store values, there will be a lock icon with the selected variable.  

To store a result:

1.  Execute operation or recalling a calculated stat variable.

2.  Press [VARIABLE].

3.  Select variable, press [ OK ].

4.  Select Store, press [ OK ].

To edit a variable's value:

1.  Press [VARIABLE].

2.  Select variable, press [ OK ].

3.  Select Edit, press [ OK ].

4.  Enter the new value, press [ OK ].

To recall a variable in a calculation:

1.  Press [ SHIFT ].

2.  Press any of the following keys to get the variable:

[ 4 ]:  A

[ 5 ]:  B

[ 6 ]:  C

[ 1 ]:  D

[ 2 ]:  E

[ 3 ]:  F

[ 0 ]:  x

[ . ]:  y

[ x10^ ]:  z

There is no ALPHA key as it was in past Casio calculators.  

[ f(x) ]/[FUNCTION]:  

Here is where we can store and use up to two functions: f(x) and g(x).  The great news is that f(x) and g(x) are no longer limited to the Table, they can be used in other modes such as Calculate.   The equations f(x) and g(x) are retained while switching modes but not retained when the calculator either is turned off or the Input/Output setting is changed.  A missed opportunity for the latter.

A plus is that either f(x) or g(x) can be a composite function.  That is f(x) can contain g(x) or g(x) can contain f(x).  

There are two ways to access the x variable:  its own key [ x ] or the key sequence [ SHIFT ] [ 0 ].  

[book]/[CATALOG]:  This is where all the functions and commands can be accessed.  The menu order of the catalog changes depending on the mode used.  Some submenus include:

Function Analysis:  The calculus functions, that used to be on the keyboard, are now stored in the Function Analysis menu:  derivative, integral, summation, logarithm*, log*, ln* (* also on the keyboard)

Probability:   % (divides the number by 100), factorial, permutation, combination, random number, random integer

Numeric:  Absolute Value, Round Off (round the number to the Fix settings internally)

Angle/Coord/Sexa...:  angle units (degrees, radians, grads, degrees minutes seconds*), polar/rectangular conversions (* also on the keyboard)

Sci Constants:  47 scientific Constants

Unit Conversions

[ three circles ]/[ TOOLS ]:  The TOOLS menu changes dynamically based on the mode.   For example, the Calculate mode will have an Undo function.

[ FORMAT ]:  Instead of the [S<=>D] key, we have the [FORMAT] key, which asks how to change the value:  Standard, Decimal, Improper Fraction, Mixed Fraction,  ENG (Engineering) Notation, Sexagesimal (degrees-minutes-seconds).

In Base mode, the [ FORMAT ] key toggles between the four bases (decimal, hexadecimal, binary, octal).

I did three comparison speed tests between the fx-991CW and fx-991EX  here:  https://www.youtube.com/watch?v=cj0Odnv0Mwk

As I understand, the fx-991CW has a faster processor than the previous fx-991EX.

Now let's talk about what is subtracted in this update.  You read this correctly, several features did not make it from the fx-991EX to the fx-991CW:

The  CALC feature where we could type in a formula, press [ CALC ] and have formula evaluation.  I was not able to find the CALC feature on the fx-991CW.

There is no longer the independent memory M, nor the storage arithmetic functions M+ and M-.  I miss this feature the most.  I really wish scientific and graphing calculators in general embrace store arithmetic like Hewlett Packard and Swiss Micros.

There is no longer the ability to use multi-statement expressions, with each expression separated with a colon.  Even when it was available, (1) storing results immediately terminated the expression (forcing the use of Ans and when available, PreAns to make using results in the next part possible), and (2) when replayed, the statement was broken up into separate parts.  

Addendum 

Note (4/12/2023):  I am wrong when I said that storing results in multi-statements on the fx-991EX (not the fx-991CW) is impossible.  We use the equals key ([ALPHA] ( = )), as in this example:

A = 9 : B = 8 * A

Csaba Tizedes uses the multi-statement and CALC feature on the fx-991EX to create an IF-THEN-ELSE structure.  In this video Tizedes uses this structure to solve equations using the Bisection method.  The IF-THEN-ELSE structure tests whether a number is positive or negative.  Please take a look his video:

https://www.youtube.com/watch?v=umxScZL1V6A

Gratitude to Csaba Tizedes, this video is shared with his permission.  

Final Thoughts

Overall, the fx-991CW is pretty solid calculator with a readable screen and a lot of features.  My favorite part of the updated is the ability to use the functions f(x) and g(x) outside of the table function.  I also like the catalog key and the page up/page down key.

I get that Casio is going for a simpler, non-busy keyboard.  However, I prefer a separate ALPHA key, along with a STORE key: it's the most efficient way to store variables.   A consequence of a non-busy keyboard is that a lot of the commands can only be accessed through menus.  The ALPHA key could have freed nine keys for more common shifted keys such as x!, polar/rectangular conversions, etc.   I would have liked to see a couple of customizable keys where we can store commands, which could be the shift of the multiplication and division keys.  The financial calculator FC-200V has two slots to store commands.

The faster processor and the better screen are pluses.  You will still get a lot for the money spent on the fx-991CW. 

Caveat:   However, if you want, need, or require subtracted features described above (CALC feature, independent memory, multi-statement expressions), you are better off buying the former fx-991EX or a fx-115ES Plus 2nd Edition.  

Unfortunately, removing the multi-statement, Casio sapped it's algorithmic power with the fx-991CW, and it needs to come back.  Furthermore, please put integer part, fraction part, and sign functions.   

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Casio fx-CG 50: Double the Investment, Rule of 72, and Rule of 69

Casio fx-CG 50: Double the Investment, Rule of 72, and Rule of 69

Introduction

Are you familiar with the Rule of 72?   In finance, the Rule of 72 refers to a simple formula to determine approximately how many years it takes an investment to double in value.  

The Rule of 72 is an approximate formula, stated as:

n = 72 ÷ I

We can determine the actual time value of money formula (without periodic payment):

FV = PV × (1 + I ÷ 100)^n

where:

FV = future value

PV = present value

I = interest rate

n = number of years

Let PV = 1 and at double the investment, FV = 2.

2 = (1 + I ÷ 100)^n

Solving for n:

ln 2 = ln [ (1 + I ÷ 100)^n ]

ln 2 = n × ln [ (1 + I ÷ 100) ]

n = ln 2 ÷ ln [ (1 + I ÷ 100) ]

This is an exact calculation.  

There is another approximation, called the Rule of 69, and the approximation is this:

n = 69 ÷ I + 0.35

In Summary:

Exact Calculation:  n = ln 2 ÷ ln [ (1 + I ÷ 100) ]

Rule of 72:  n = 72 ÷ I

Rule of 69:  n = 69 ÷ I + 0.35

Comparison

A spreadsheet, generated using a Casio fx-CG 50, is shown for rates from I = 2% to I = 20% in increments of 1%.

The Rule of 72 is the most accurate at I = 8%, with an error of 6.4 × 10^-3.  At interest rate I ≈ 7.846871453, the Rule of 72 equals the exact calculation.    

When I < 7.846871453, the Rule of 72 over-estimates.  Other wise the Rule of 72 under-estimates.  

For I = 2 to at least I = 20, the Rule of 69 under-estimates, but the difference shrinks as I increases.  

Here is a graphical comparison for values of I = 5% to I = 15%.  

Source

Thomsett, Michael C.  The Real Estate Investor's Pocket Calculator 2nd Ed. AMACOM:  New York.  2018.  ISBN 9780814438893 (paperback)

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Casio Comparison: fx-9750GII vs. fx-9750GIII

Casio Comparison:  fx-9750GII vs.  fx-9750GIII

Casio claims the fx-9750GIII is comparable to the TI-84 Plus.  But what about its predecessor: fx-9750GII? 

Casio fx-9750GII

Screen: Casio fx-9750GII

Case and Back: Casio fx-9750GII

Casio fx-9750GIII

Screen: Casio fx-9750GIII

Case and Back: Casio fx-9750GIII

In the United States, we have a new graphing calculator from Casio, the fx-9750GIII.  It is an upgrade of the fx-9750GII in several ways:

*  A new Python programming mode
*  The spreadsheet and geometry modes are added
*  Storage memory of 3 MB is allocated to hold geometry, python, and flash files.  This is an addition to the about 61K of regular memory.
*  The fx-9750GIII now has textbook input and output options.   In textbook mode, you can control whether output defaults to exact answers (fractions, square roots, terms of pi) or approximate answers.

The fx-9750GIII's case is slightly smaller than the fx-9750GII, but the width is the same.  The fx-9750GIII has a lighter body as well.  I really like the lighter feel of the fx-9750GIII.  The fx-9750GIII's back has a 3D feel to it, which is nice.  I am also impressed with the keyboard and the color contrast. 

The package I purchased did not have a USB cord.   I believe the proper cord is a mini USB cord, probably can use the cord that came with the fx-9750GII (and similar calculators) and it should be fine. 

Performance wise, the two calculators are about even, with the fx-9750GIII slightly faster. I only did a little bit of testing so far.   Neither of them are slouches. 

For a more detailed comparison, you can download a PDF comparison here:
https://drive.google.com/open?id=1bMC27Mbjw4N41qv8CZ1UD5qTJopbAVZC

(PDF document created with LibreOffice Writer)

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.