Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
ARITHMETIC and GEOMETRIC SEQUENCES
Prior to this course, you should have studied two special sequences, arithmetic and geometric.
Review your notes and the problem sets, if you have them, talk with the other students in your
group and then write out the definition of an arithmetic sequence. What is the defining
characteristic of an arithmetic sequence?
Similarly, write out the definition of a geometric sequence. What is the defining characteristic
of a geometric sequence?
1.
Label each of the following sequences as either Arithmetic, Geometric, neither, or both.
a.
2, 7, 12, 17, ...
b.
2, 4, 7, 11, 16, ...
c.
0.5, 1, 2, 4, 8, ...
d.
1 1 1 1
, , , ,
12 6 4 3
e.
, , , ....
f.
1, 1, 2, 3, 5, 8, ...
Write out several terms and then determine the type of sequence.
g.
Starting with $5000, the amount of
money in a CD at the end of each month
if it earns 6% annual interest,
compounded monthly.
h.
An item costing $625 is bought on layaway. $175 is made as a down payment
and $50 is paid each month. The amount
of payment remaining each month.
Seq & Ser. 3.1
Rev. S06
Mathematical Investigations IV
Name
Again, consider arithmetic sequences. Express each of the following both recursively and
explicitly.
2. 3, 7, 11, 15, ...
3. 20, 13, 6, –1, ...
4.
 _______________ if n  1

an = 
 _______________ if n  1
 _______________ if n  1

an = 
 _______________ if n  1
an =
an =
Generalize: Let an  be an arithmetic sequence. Write out the next few terms.
a1 = a,
5.
a2 = a + d,
a3 =
,
a4 =
,
a5 =
Write out a recursive definition for an arithmetic sequence:
 _______________ if n  1

an = 
 _______________ if n  1
Write out an explicit definition:
an =
Let's repeat the work above with geometric sequences. Express each of the following both
recursively and explicitly.
6. 3, 15, 75, 375, ...
8.
7. 12, 6 2 , 6, 3 2 , ...
 _______________ if n  1

an = 
 _______________ if n  1
 _______________ if n  1

an = 
 _______________ if n  1
an =
an =
Generalize: Let an  be a geometric sequence. Write out the next few terms.
a1 = a,
a2 = ar,
a3 =
,
a4 =
Seq & Ser. 3.2
,
a5 =
Rev. S06
Mathematical Investigations IV
Name
Seq & Ser. 3.3
Rev. S06
Mathematical Investigations IV
Name
9.
Write out a recursive definition for a geometric sequence:
 _______________ if n  1

an = 
 _______________ if n  1
Write out an explicit definition:
an =
10 – 13. Write a formula for the nth term and then find the value of the requested term.
10.
Find the 53rd term of the arithmetic sequence 8, 11, 14, 17, …
11.
Find a83 for the arithmetic sequence 2, 1.4, 0.8, 0.2, …
12.
Find a5 for an arithmetic sequence if a1 = 3 and a23 = 38.2.
13.
Find a14 for the geometric sequence 12, 8,
16
, ...
3
Seq & Ser. 3.4
Rev. S06