1. Sequences and Series
It’s all in Section 9.4a!!!

2. “sequence” will refer to
Sequence – an ordered progression of numbers – examples: 1.
Finite Sequence 2.
Infinite Sequences 3.
(unless otherwise
specified, the word
“sequence” will refer to
an infinite sequence) 4.
which is sometimes abbreviated
Notice: In sequence (2) and (3), we are able to define a rule for
the k-th number in the sequence (called the k-th term).

3. Practice Problems
Find the first 6 terms and the 100th term of the sequence
in which
Note: This is an explicit rule for the k-th term

4. Practice Problems
Another way to define sequences is recursively, where we
find each term by relating it to the previous term.
Find the first 6 terms and the 100th term for the sequence defined
recursively by the following conditions:
for all n > 1.
The sequence:
The pattern???

5. Definition: Arithmetic Sequence
A sequence is an arithmetic sequence if it can be written
in the form
for some constant d.
The number d is called the common difference.
Each term in an arithmetic sequence can be obtained recursively
from its preceding term by adding d:
(for all n > 2).

6. Practice Problems
For each of the following arithmetic sequences, find (a) the
common difference, (b) the tenth term, (c) a recursive rule for the
n-th term, and (d) an explicit rule for the n-th term. 1.
(a) The difference ( d ) between successive terms is 4.
(b)
(c)
(d)

7. Practice Problems
For each of the following arithmetic sequences, find (a) the
common difference, (b) the tenth term, (c) a recursive rule for the
n-th term, and (d) an explicit rule for the n-th term. 2.
(a) The difference ( d ) between successive terms is –3.
(b)
(c)
(d)

8. Practice Problems
For each of the following arithmetic sequences, find (a) the
common difference, (b) the tenth term, (c) a recursive rule for the
n-th term, and (d) an explicit rule for the n-th term. 3.
Is this sequence truly arithmetic???
Difference between successive terms:
We do have a
common difference!!!

9. Practice Problems
For each of the following arithmetic sequences, find (a) the
common difference, (b) the tenth term, (c) a recursive rule for the
n-th term, and (d) an explicit rule for the n-th term. 3.
(a) The difference ( d ) between successive terms is ln(2).
(b)

10. Practice Problems
For each of the following arithmetic sequences, find (a) the
common difference, (b) the tenth term, (c) a recursive rule for the
n-th term, and (d) an explicit rule for the n-th term. 3.
(c)
(d)

11. Geometric Sequences

12. Definition: Geometric Sequence
A sequence is a geometric sequence if it can be written in
the form
for some non-zero
constant r.
The number r is called the common ratio.
Each term in a geometric sequence can be obtained recursively
from its preceding term by multiplying by r :
(for all n > 2).

13. Guided Practice
For each of the following geometric sequences, find (a) the
common ratio, (b) the tenth term, (c) a recursive rule for the n-th
term, and (d) an explicit rule for the n-th term. 1.
(a) The ratio ( r ) between successive terms is 2.
(b)
(c)
(d)

14. Guided Practice
For each of the following geometric sequences, find (a) the
common ratio, (b) the tenth term, (c) a recursive rule for the n-th
term, and (d) an explicit rule for the n-th term. 2.
(a) Apply a law of exponents:
(b)

15. Guided Practice
For each of the following geometric sequences, find (a) the
common ratio, (b) the tenth term, (c) a recursive rule for the n-th
term, and (d) an explicit rule for the n-th term. 2.
(c)
(d)

16. Guided Practice
For each of the following geometric sequences, find (a) the
common ratio, (b) the tenth term, (c) a recursive rule for the n-th
term, and (d) an explicit rule for the n-th term. 3.
(a) The ratio ( r ) between successive terms is –1/2.
(b)
(c)
(d)

17. Guided Practice The second and fifth terms of a sequence are 3 and 24,
respectively. Find explicit and recursive formulas for the
sequence if it is (a) arithmetic and (b) geometric.
If the sequence is arithmetic:
Explicit Rule:
Recursive Rule:

18. Guided Practice The second and fifth terms of a sequence are 3 and 24,
respectively. Find explicit and recursive formulas for the
sequence if it is (a) arithmetic and (b) geometric.
If the sequence is geometric:
Explicit Rule:
Recursive Rule: