Intensive Algebra 2
Name______________________
Intro to Arithmetic & Geometric Sequences
Sept. 9
Arithmetic Sequences:
Arithmetic sequences have the form:
a, a  d , a  2d , a  3d , a  4d ,...
where, only in terms of a (the first term), an

Let’s try some non-trivial examples…
Ex1: The 12th term of an arithmetic sequence is 32 and the 5th term is 18. Find the 20th term.
Ex2: Which term of the arithmetic sequence 1, 4, 7, 10,... is 88?
Partial Sums (Arithmetic sequences):
Recall the concept of a partial sum:
S1  a1
S2  a1  a2
S3  a1  a2  a3
S4  a1  a2  a3  a4
Suppose an is an arithmetic sequence.
Then, only in terms of a1 and d the nth partial sum is given by…
Sn 
Intensive Algebra 2
Proof of Finite Sum of an Arithmetic Sequence:
When asked to prove a mathematical statement (often presented in the form of a formula), we
need to make sure we are presenting an argument that is clear, organized, and efficient. There
are various proof techniques in mathematics. We will practice the art of performing proofs
over the next few days, and discuss the merits of some of the techniques used by
mathematicians.
Let’s look at the following exercise…
Given: an is an arithmetic sequence
Prove: ___________________________________________ (from conclusion on previous exercise)
Proof:
Intensive Algebra 2
Let’s try an application of the principle for the nth partial sum of an arithmetic sequence…
Ex1: In their free time, Calvin & Hobbes love to get involved with new projects. Their most
recent project involves building a cool looking object with uniform snowballs. They want to
construct a figure resembling an isosceles trapezoid with the first row of snowballs containing
12 snow balls, the second row 15 snowballs, the third row 18 snowballs, …. How many rows of
snowballs will be needed if Calvin & Hobbes want to use all 882 snowballs they have packed?
Intro to Geometric Sequences:
Arithmetic sequences are one in which consecutive terms have a common ________________.
Geometric sequences are one in which consecutive terms have a common _________________.
The following is an example of a geometric sequence:
3, 6,12, 24,...
Derive a geometric sequence whose terms have alternating signs and whose absolute values
are decreasing.
A geometric sequence is a sequence in the form…
a, ar, ar 2 , ar 3, ar 4 ,...
where, only in terms of a and r (the first term), an

Intensive Algebra 2
Partial Sums (Geometric sequences):
Suppose an is a geometric sequence.
Then, only in terms of a1 , the nth partial sum is given by…
Sn 
Proof of Finite Sum of a Geometric Sequence:
Given: an is a geometric sequence
Prove: ___________________________________________ (from conclusion above)
Proof:
Intensive Algebra 2
Here are some nice little challenges…
Ex1: Find the partial sum, S n , of the geometric sequence that satisfies the given conditions:
a6  224
a3  28
n6
Ex2: Find the sum of the series 1  2  4  5  7  8  10  11  ...  299 , which is the sum of all
integers except for multiples of three. Be sure your approach is clearly communicated.
Infinite Sums
Using some thought and intuition, if we add an infinite number of terms of an arithmetic
sequence, the sum is either ________ or _________. This should be fairly clear to you!
Suppose we look at a geometric sequence and add an infinite number of terms.
Q: What condition must be true of the common ratio, r, in order for an infinite geometric series
to converge? ______________________________________
Suppose an is a geometric sequence, with __________________
Then,

an 

n 1
Ex: Determine the infinite sum, should it exist (first determine if this series converges).
2 4
8
16
 

...
5 25 125 625