1
SECTION 8.1
SEQUENCES
A
sequence
is an ordered list of numbers:
a
1
,
a
2
,
a
3
, …
Notation:
Sequences may bedefined by giving a formula for the
n
’th term
:
Ex. 1a.
0
1
2
n
n
∞
=
a
n
=
1
2
n
First four terms:
Ex. 1b.
1
3
21
n
n
n
∞
=
+
a
n
=
3
n
n
+
First four terms:
Sequences may also be defined recursively
Ex 1c. The Fibonacci sequence {
f
n
}:
f
1
= 1,
f
2
= 1,
f
n
=
f
n
– 1
+
f
n
– 2
for
n
≥
3.
Write out the first 12 terms of the Fibonacci sequence
Ex. 1c. Write out the first 6 terms of the sequence
0
sin
4
n
n
p
∞
=
.
Graphing Sequences using the TI-89/92
(TI-82, -83, calculators will do this too.
For TI-86, see note on page 568.)
Press
MODE
and change the graph mode to
SEQUENCE
, press
ENTER
twice.
Press (
green )
Y=
, and observe that the entries in this menu appear in pairs: u1, ui1,u2,ui2,
etc. and that the variable for sequences is
n
.
You enter the expression for the term in terms of
n
in the u’s.
If your sequence is given recursively
(the value of a term depends on previous terms),
you’ll need to give the first
term on the ui line.
(It’s strange notation!)
We’ll plot the first 20 terms from the sequences of example 1a and 1b.
On the command line
enter
u1(n) = 1/2^n
. It is not necessary to use ui1 since 1/2^n is an explicit formula.
Next, enter
u2(n) = 3*n/(2*n+1);
again ui2 is not used.
Press (
green )
WINDOW
and set the values as follows:
nmin = 0,
nmax = 20,
plotstrt = 1,
plotstep = 1, xmin = 0,
xmax = 20, xscl = 1, ymin = 0,
ymax = 2, yscl = .2
Lastly, press (
green )
GRAPH
and you’ll see the lower set of dots representing the numbers
of the sequence
0
1
2
n
n
∞
=
approaching
, and the upper set representing the sequence
3
n
n
+
approaching the value
.
You can see the values by pressing
F3
(
Trace
) and moving the cursor to the right or up and
down between sequences.
You CAN move the cursor to the left, but it is v-e-r-y slow!