Investigative Skills Toolkit (CAS)
Student Task Sheet – TI-Nspire CAS Version
Introduction
This activity will develop your ability to use the TI-Nspire CAS platform to investigate and
generalise results from numerical patterns.
The TI-Nspire and Mathematical skills that will be developed are:
 defining functions
 using sigma notation
 generating sequences
 using statistical regression models to reveal algebraic generalisations.
Mathematical Aim
As a context for developing the above 4 skills, you will use them to find the algebraic formula for
the sum:
1+2+3+4+....+n
You will then find the algebraic formula for the sum:
12+22+32+42+....+n2
and finally find the formula for:
13+23+33+43+....+n3
Note: There are many ways of going about deriving the formula for the sequence sums that are
listed above. Some of these ways are quicker than the approach you will follow here. However,
an important purpose of this task is to have you experience how a wide variety of technological
skills can be brought together to work in powerful ways. Such techniques are able to be used
even when the formulae being sought are not as well known as the above three sequence sums.
Structure of Tasks
Tasks 1, 2, 3 and 4 introduce each of the 4 individual skills, listed above.
Task 5 merges these skills together in order to acheive the Mathematical Aim given above.
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Task 1 – Defining Functions
A function has three things: a name, an input and an output.
Create a New Document and insert a Calculator Page.
Type in g(x):=x+2 and press ·. as shown on the right.
Notice the colon in front of the equals sign.
You can quickly obtain := by pressing / then t.
This line reads as “g of x is defined as equal to x plus two”
Now type g, and you will see that it is already displayed in
bold. This means that the Nspire system knows that
something named as g already exists in the current problem.
Now continue to type in g(4) and press ·. You should be given the answer of 6. The
function g took the input of 4, and after adding 2 to it, presented the output of 6.
Another way of accessing the function of g is to press the h key. Do that now. Nice that it
includes the brackets for you, isn’t it?
Now type g(-7) and before you press ·, see if you can correctly predict what it will display.
Using an Nspire CAS handheld, you can input more than just numbers to your function:
Use the h key to type g(t) and press ·.
Then type g(n) and press ·.
Then type g(banana) and press ·.
You will see that the function g takes whatever it is given as an input, and generates an
output that is simply the input plus two.
Add a new calculator page by pressing / then ~ then ‘Add Calculator’
Define a new function called h(x,y):=x+y2, as shown on the
right.
This function has two inputs, and one output.
Use the h key to type h(2,3) and before you press ·,
see if you can correctly predict what it will display.
Repeat for h(5,-4).
Then predict what you think h(r,r) will give, then check it.
Watch carefully what happens when you type in h(g,g) and be sure that you know why it
gives the error message that it does when you press ·.
Ask your teacher if you are unsure.
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Add a new calculator page by pressing / then ~ then ‘Add Calculator’
Define a new function called discrim, as shown below
discrim(tom,dick,harry):=dick2- 4·tom·harry
This function has three inputs, and one output, and the
variable names are no longer single letters. Notice the
multiplication dots between 4 and tom and harry.
Use the h key to type discrim(1,2,3) and before you press
·, see if you can correctly predict what it will display.
Repeat for discrim(5,-2,7).
Then predict what you think discrim(a,b,c) will give, then check it.
Finally, type discrim(p,5p,p+1) and remember to predict the output before you press ·.
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Task 2a – Sigma Notation
Σ is the Greek capital letter called ‘sigma’. A lowercase letter sigma looks like this: σ
σ is used in Statistics to represent a measure of how spread out a data set is.
Σ is used to mean the adding up of several terms. It requires three ‘inputs’: a lower bound, an
upper bound and an algebraic expression.
Here are two examples, which you shall shortly check on the Nspire. Don’t type anything in yet.
Example (i).
5
 (r ) , which reads as “the sum from r equals 1 to 5 of r”
r 1
This is a short-hand way of writing 1+2+3+4+5, which is equal to 15.
Example (ii).
9
 (c
2
) , which reads as “the sum from c equals 6 to 9 of c squared”
c6
This is a short-hand way of writing 62+72+82+92, which is equal to 230.
In each example expression, the letter under the sigma sign increases by one each time from
the lower bound until it reaches the upper bound. For each value it takes on this journey, the
algebraic expression is evaluated.
You will now check the two results for examples (i) and (ii):
Insert a new Problem by pressing:
~ ... Insert ... Problem ... Add Calculator.
To obtain the sigma notation template, press t and select
the sigma notation icon, as shown on the right.
You will then have an empty template as shown, and the
cursor is flashing in the region where you need to type in the
variable will appear in your expression.
Set about entering in the letters and numbers as shown in
example (i), above, and then press ·.
Repeat for example (ii).
continued..../
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
You should now have a screen display like this:
Now edit one of these expressions to generate the sigma
notation to find the following sum:
22+32+42+....+72
You should obtain the answer of 139.
Can you now predict the values of each of the following summations, before you press ·?
So, to finish, sigma notation can be used very effectively to add up lots of terms in a sequence
for which you already know an algebraic expression for each term.
You can consider the sigma notation to be a function with 3 inputs: the lower bound, the upper
bound and a formula. It generates a single output of a value, or an expression, for the sum of all
the terms it represents.
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Task 2b (Extension) - Sigma Notation Equations
You can use sigma notation with other Nspire functions....
Consider the problem of determining the value of n such that
1+2+3+4+....+n = 153.
You need to solve for n, which may at first glance seem only
to be able to be done by a method of trial and improvement.
As can be seen on the right, both the nSolve(...) and
solve(...) commands can be used. Find these by pressing:
b ... Algebra ... Solve
or b ... Algebra ... Numerical Solve
Notice when using the solve(...) command, a second solution for the upper bound is also
given. You can discard this, as –18 is lower than the lower bound of 1. Alternatively, you
might want to consider why –18 is considered by TI-Nspire CAS to be a possible solution.
Consider another problem of determining the value of p such
that 5p+6 p+7 p+8 p = 1196.
Before you look at the screenshot solution on the right, can
you set up an nSolve(...) or solve(...) statement to obtain the
value of p?
Consider a final problem of finding out the sum of which 40
odd numbers equals a total of 2640.
Before you look at the screenshot solution on the right, can
you set up an nSolve(...) or solve(...) statement to obtain the
answer to this puzzle?
Hint: the 40th odd number after the start is 39 odd numbers
after the first odd number!
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Task 3 – Generating Sequences
The sequence command can generate a sequence of terms, based upon a formula, a start value
and end value. The default setting is that it increases its variable by one each time.
You can hopefully appreciate that it has very similar syntax to the sigma notation.
However, the sigma notation sums all terms in a sequence without letting you see them.
The sequence command will show you all the terms, but not sum them.
Insert a new Problem by pressing: ~ ... Insert ... Problem ... Add Calculator.
The first example (left) generates the sequence of squared
numbers from 12 to 42.
The second example generates the sequence of cubed
numbers from 43 to 63.
Notice that the output of this command is not a single value,
but a list of values, as denoted by the ‘curly’ brackets, {...}.
The syntax of the command is: seq( formula, variable in formula, start value, end value )
Type in the following commands, and see if you can predict what they will display, before you
press ·.
seq(4-k,k,2,7)
seq(5a+3,a,19,25)
seq(ca,a,2,6)
Now try to type the sequence commands that will display each of the following outputs:
1. the first five odd numbers
2. the sequence {75, 69, 63, 57, 51}
3. the sequence {4y5, 5y6, 6y7, 7y8}
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Task 4 – Using Statistical Regression Models
When you obtain a sequence of numbers and you are not sure what formula generated them, it
is often helpful to plot them on a graph.
Shown below are typical graphs of the most commonly occuring types of formulae
formula with n2 in it
formula with n3 in it
formula with n4 in it
As you can see, there are only subtle differences between them, so a bit of intelligent decision
making is required when trying to apply these formulae.
For a worked example, you shall create a sequence using a formula that you already know, plot
it, and then use Statistical Regression to provide you back with the formula you used. This will
act as a ‘self-check’ process, in readiness for when you are using the technique to find an
unknown formula.
Insert a new Problem by pressing: ~ ... Insert ... Problem ... Add Lists & Spreadsheet.
Then type into the very top left cell on the screen, above the
greyed-out row the variable name “n.value” and press ·.
Then move to the very top of column B, and type in
“term.value” and press ·.
Notice that you can use the decimal point as a valid character
in the names of our variables, and it helps to make them
more readable.
You shall now generate the first 10 terms in the sequence whose nth term is given by 4n2-3n
As ever, there are many ways to do this.....
First, you shall generate the n.values from 1 to 10.
You can use the sequence command for this:
seq(n,n,1,10)
To enter this command, move the cursor to the grey row,
second from the top, in column A, and press =.
The screen will then look like that shown on the right:
Type in seq(n,n,1,10) and press ·.
You should then see column A fill up with the numbers 1 to
10.
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
You shall repeat this process for column B, only this time the
sequence command will be:
seq(4n2-3n,n,1,10)
When you have done this, your screen should look like this:
Now insert a new page with Data & Statistics, and click on
where it says ‘Click to add variable’ at the foot of the screen.
Select n.value
Move to the left of the screen, where it (used to) say ‘Click to
add variable’ and click again.
Select term.value
You should then have a screen looking like the one shown on
the right.
You already know that the formula for this sequence is a
quadratic expression, but you would be forgiven for thinking it
might be cubic or quartic if you didn’t already know.
Look back and compare this graph with the 3 screenshots at
the very start of Task 1d.
You will now instruct the Nspire to fit the best formula it can to
these plotted points.
Press b ... Analyze ... Regression ... Show Quadratic
The regression formula will then be displayed as shown on
the left.
You were expecting 4n2-3n and it has displayed something
very similar to this, only with different variable names and
surplus characters. This is to be expected when using
Regression techniques.
Move the cursor over the displayed equation and press /
then b then select ‘Remove Regression’
If you didn’t know it was a Quadratic formula that you were
after, you could have done:
b ... Analyze ... Regression ... Show Cubic
or even
b ... Analyze ... Regression ... Show Quartic
Do each of these now.
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
You will notice that when selecting ‘Show Quartic’, the displayed regression equation seems
overly complex.
The full equation looks like this (split up so that you can see each term more clearly):
y= −1.4648437500019E −13·x4
+3.2226562500042 E −12·x3
+3.9999999999767·x2
-2.999999999939·x
-4. E −11
or, in a shortened form,
y= −1.464E −13·x4+3.222 E −12·x3+3.999·x2 -2.999·x -4. E −11
You can see something like y=4·x2 -3·x lurking in the middle of this expression, albeit in a slight
disguise due to rounding errors from the regression process!
And you have surplus terms which are very small indeed.
For example, look at the coefficient of the x4 term.
It is −1.464E −13
This is the TI-Nspire’s way of displaying the number −1.464×10−13
Which is the number –0.0000000000001464
You can therefore conclude that the x4 term is not there as its coefficient is such a small number.
Similarly for the coefficient of x3 which is 0.000000000003222
And for the constant on the end, which is –0.00000000004
This leaves you with y= 3.999·x2 -2.999·x
You can take this to mean y=4·x2 -3·x after ‘adjusting’ for rounding errors.
It is very important that you understand the above logical process, else you may make errors
when interpreting results from Statistical Regression calculations.
You can, of course, have the TI-Nspire CAS help you with this interpretation process!
You can instruct the regression formula to be displayed with slightly reduced accuracy, thereby
revealing the most important terms.
To do this, insert a new Calculator Page.
Press h and select stat.regeqn
Note: all the other variables that start with stat. were
automatically generated by the Statistical Regression process
that you commanded it to perform.
Then type the letter x and close the brackets, so that the line
reads as stat.regeqn(x)
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Then press:
b ... Number ... Approximate Fraction
You will have a command line that reads as:
stat.regeqn(x)▶approxFraction(5.E−14)
Don’t press · yet!
Edit the last characters to change it from –14 to –10:
stat.regeqn(x)▶approxFraction(5.E−10)
This instructs it to start rounding at about the 10th decimal
place, rather than the 14th decimal place.
The full answer that you will obtain is shown above, but you will have to scroll left and right to
read it on the handheld.
You can now see more clearly the desired formula of y=4·x2 -3·x
You can also see the other terms which have such large denominators that they can be
ignored, as their contribution will be so small.
Statistical Regression Conclusion
Statistical Regression is very powerful, but it requires careful interpretation to draw the correct
conclusions.
It is also worth remembering that its output is not proof that the formula works.
It is merely the best approximation given the data that was processed and the type of regression
equation that you were trying to fit to that data.
Task 4b - Statistical Regression Practice
The previous example was based around the formula for the nth term to be 4n2-3n.
You will now change the setup so that the TI-Nspire system tries to detect the formula 5n3-n/7
Return to the Lists & Spreadsheet page and edit the seq(4n2-3n,n,1,10) command in column B
to read seq(5n3-n/7,n,1,10)
All the other dependant pages will automatically update, and the new regression equation will be
displayed on the graph.
On the Calculator page you should re-execute the command line:
stat.regeqn(x)▶approxFraction(5.E−10)
If this does not obviously reveal the expression 5x3-x/7, then you will have to adjust the tolerance
of the approxFraction command from 5.E−10 to something like 5.E−8 or 5.E−6
This adjustment is because of the fraction 1/7 that is in the original formula.
This is further evidence that you must remain ‘flexible’ when using Statistical Regression, as the
complexity of the formula that you are seeking often affects the accuracy of the calculations.
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Task 4c - Find the Formula
Tasks 4a and 4b were built around already knowing the generating formula.
This is not really realistic, as why would you try to find out something you already knew?!?
Consider the following sequence:
2, 14, 50, 130, 280
Your task will be to find a formula for the nth term.
To do this, return to the Lists & Spreadsheets page and do
the following:
1. move to the grey formula row of column B and press . to
delete the sequence statement.
2. with the selected cell still in the grey formula row of
Column B, press / then b then select Clear Data
You should then have a screen similar to that shown on the
right.
Now, type in the 5 sequence values into column B, to give a
screen similar to that shown on the right:
You may have noticed that column A still has a sequence
running from 1 to 10, but you do not have 10 numbers in
column B.
You will need to edit the formula in column A to generate
only the numbers from 1 to 5.
Do that now.
Move back to the Data & Statistics graph to verify the
plausibility of fitting a quartic, cubic or quadratic equation to
the data.
You can choose to fit whichever type of equation you think
best suits.
Remember that if you opt for, say, a quartic, and the data is
only quadratic, then you should expect the coefficients of the
x4 and x3 terms to be neglible.
However if you opt for a quadratic and the data is quartic, then the regression line will not fit the
data well and the coefficients in the equation will most likely be ‘ugly’
Once you have completed this process, and chosen the
required amount of accuracy for the approxFraction
command, compare your result to the screen on the right.
The formula still looks unappealing with all the fractional
coefficients, but type in the factor(...) command as shown
and press · to see a more elegant form of the formula.
You should now check that this formula does indeed generate the first 5 terms that you were
given - why not use the factorised form that you have just created, and work through the
evaluation process on paper, without using the TI-Nspire to help you!
To practice the above process, edit the data on your TI-Nspire to help you identify the formulae
for the nth term for each of the following two sequences:
a) 0, 4, 22, 70, 170
b) 0, 3, 11, 26, 50
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Task 5a – Combining Formulae + Sigma Notation + Sequences +
Statistical Regression
Insert a new Problem by pressing:
~ ... Insert ... Problem ... Add Calculator.
Enter both of the statements as shown on the right, which
define a variable called power and a function called total.
As the variable power equals 1, if you were to enter in
total(5), it would calculate the sum 1+2+3+4+5.
Try it now.
You will be able to change the value of power to other numbers later on, and therefore all the
calculations that you will design in the next few steps will be automatically updated.
Now define a variable called terms:=6 (see right)
Entering the command total(terms) ought to calculate the
sum 1+2+3+4+5+6.
Try it now.
You are now going to use all the techniques learnt so far to find the formula for
1+2+3+4+5+...+n
1. Insert a new Lists & Spreadsheet page
2. Type into the very top left cell on the screen, above the greyed-out row the variable name
“n.value” and press ·.
3. Move to the very top of column B, and type in “term.value” and press ·.
4. Move the cursor to the formula row in column A, and press =.
5. Type in seq(r,r,1,terms) and press ·.
6. Move the cursor to the formula row in column B, and press =.
7. Type in seq(total(r),r,1,terms) and press ·.
You should have a screen that now looks like the one given below:
The purpose of using the variable terms, makes sure that
both lists have the number number of terms in them.
You should now:
1. insert a Data & Statistics page
2. plot the data
3. fit an appropriate regression line
4. insert a Calculator page
5. if required, use the approxFraction command to extract
the desired formula from the regression equation
(stat.regeqn)
© 2011 Nevil Hopley and Texas Instruments Education Technology
Investigative Skills Toolkit (CAS)
Compare your results to the screenshot on the right.
By using the variable n, you can more quickly see the formula
for 1+2+3+4+5+...+n
As before, the factor(...) command tidies up the result nicely.
As you are using the TI-Nspire CAS handheld, you can
compare this result to the one generated by the in-built
algebra system, by typing total(n).
This last step may appear to have completely negated everything that has been covered in this
whole activity....not so!
Were you to use a TI-Nspire non-CAS handheld, then you would have to utilise all the tools of
the TI-Nspire in the manner that you have done in this activity, as you would not have an algebra
system upon which to rely.
Task 5b – Sum of the first n Square Numbers
As declared at the outset of this whole activity, you will now attempt to find the formula for
12+22+32+42+....+n2
This requires very little extra work, as you have already constructed a fully dynamic system of
pages in the current problem.
1.
2.
3.
4.
Return to the calculator page where you defined power to equal 1.
Now redefine power:=2
This will automatically update the sequences and data plots.
You may have to try a different type of regression function (quartic/cubic/quadratic),
depending upon what you previously used.
5. Use the approxFraction and factor(...) commands to obtain the nicest form for the formula.
6. Compare and check your final answer by using the total(n) command.
Task 5c – Sum of the first n Cube Numbers
Finally, you will find the formula for 13+23+33+43+....+n3
1. Repeat the above 6 step process, redefining power:=3
Task 5d (Extension) - Sum of the first n Quartic Numbers
It is possible to attempt to find the formula for 14+24+34+44+....+n4
However, you need to try it yourself to discover what happens.
You can use the total(n) command to know what you are aiming for.
 Can you find a way of using Statistical Regression to obtain the same formula?
 Can you identify the ‘feature’ of this last challenge that causes the whole statistical
regression process on the TI-Nspire to falter?
© 2011 Nevil Hopley and Texas Instruments Education Technology