home page TI-82, TI-83 Sequences on a TI-83
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A series of activities on sequences using a TI-83
Written to complement training materials for teachers on using graphical calculators.

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4. Extending to more difficult sequences

What is the nth triangle number?



The triangle numbers are a difficult sequence to work out a formula for, so we introduce the idea of differences. Here are the 1st, 2nd and 3rd differences for the square numbers.
Term 9 16 25 36 49 36
1st difference 7 9 11 13 11
2nd difference 2 2 2 2
3rd difference 0 0 0

Notice that the second differences are constant. To set this up of your calculator, you will use lists L3, L4 and L5, and the special TI-83 function DList (found in LIST OPS 7).

Also at this stage we will enter formulas into lists in a slightly different way, using the " button (ALPHA +) to make the function "permanent". This makes the calculator act similarly to a spreadsheet.

Enter these instructions
Key presses
Enter 1, 2, 3, 4, 5... in L1
" left bracketL1 add2 right bracket square " store L2 enter
" DList left bracket L2 right bracket " store L3 enter
" DList left bracket L3 right bracket " store L4 enter
" DList left bracket L4 right bracket " store L5 enter
 

This will create the sequence 9, 16, 25, 36... in L2 and the first, second and third differences in L3, L4 and L5 respectively.
screen 9 screen 10

If the sequence in L2 is changed the differences in L3, L4 and L5 will update automatically. Choose a variety of sequences and look to see where their differences become constant. Here are some examples taken from Section 3.


Sequence Key presses nth term Constant differences
1, 3, 5 ,7, … L1 times 2 subtract 1 store L2 enter 2n-1 1st
7, 9, 11, 13… L1 times 2 add 5 store L2 enter 2n+5 1st
multiples of 7 L1 times 7 store L2 enter 7n 1st
square numbers L1 square store L2 enter n2 2nd
cube numbers L1 times L1 times L1store L2 enter n3 3rd
10, 9, 8, 7… 11 subtractL1 store L2 enter 11-n 1st
       

Some sequences have first differences constant. These are sequences whose nth term is a linear function. Other sequences - the squares for example - had second differences constant. Their nth term is a quadratic function. The cube numbers have their third differences constant. So, finding which differences are constant will tell you about the order of the function giving the nth term.

When the triangle numbers are entered in L2, the second differences are constant. So, what is the nth term formula for the triangle numbers? What is the 100th triangle number?
Triangle numbers Key presses nth term 100th term
1, 3, 6 ,10, … left bracketL1 square add L1 right bracket divide 2 store L2 enter (n2+n)/2 5050