Introduction to sequences, arithmetic, geometric and others. Recursively and implicitly definitions. Using the graphing calculator find the value of any term in a sequence.

1. Sequences
Photo source: Recursive Blanket Flower

2. Find the next three terms in each sequence of numbers ...
4, 7, 10, 13, , ,
3, 6, 12, 24, , ,
32, 16, 8, 4, , ,
1, 1, 2, 3, 5, 8,13, , ,

3. 4, 7, 10, 13, , ,

4. Arithmetic sequences on the calculator ...

5. Sequence: An ordered list of numbers that follow a certain pattern (or rule).
Arithmetic Sequence:(i) Recursive Definition: An ordered list of numbers
generated by continuously adding a value (the common
difference) to a given first term.
(ii) Implicit Definition: An ordered list of numbers where
each number in the list is generated by a linear equation.
Common Difference (d):(i) The number that is repeatedly added to
successive terms in an arithmetic sequence.
(ii) From the implicit definition, d is the slope of the linear
equation.

6. To Find The Common Difference
d = tn - t(n - 1)
d is the common difference
tn is an arbitrary term in the sequence
t(n - 1) is the term immediately before tn in the sequence
To Find the nth Term In an Arithmetic Sequence
tn = a + (n - 1)d
tn is the nth term
a is the first term
n is the quot;rankquot; of the nth term in the sequence
d is the common difference
Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ...
Solution: a = 11 t51 = 11 + (51 - 1)(-6)
d = 5 - 11 t51 = 11 + (50)(-6)
= -6 t51 = 11 - 300
n = 51 t51 = -289

7. 3, 6, 12, 24, , ,

8. Geometric sequences on the calculator ...

9. Geometic Sequence: (i) Recursive Definition: An ordered list of numbers
generated by continuously multiplying a value (the common ratio)
with a given first term.
(ii) Implicit Definition: An ordered list of numbers where each
number in the list is generated by an exponential equation.
Common Ratio (r): (i) The number that is repeatedly multiplied to
successive terms in a geometic sequence.
(ii) From the implicit definition, r is the base of the exponential
function.

10. To Find The Common Ratio
r is the common ratio
tn is an arbitrary term in the sequence
t(n - 1) is the term immediately before tn in the sequence
To Find the nth Term In a Geometic Sequence
tn is the nth term
a is the first term
n is the quot;rankquot; of the nth term in the sequence
r is the common ratio