IB SL 1 - Lesson 5.2: Geometric Sequences, Sigma Unit 5 – Series and Sequences
Notation, and Series
Big Picture
We are in our first weeks of Semester 2. We have started our Journey
into Unit 5, Series and Sequences. We will continue this for a few more
months before we begin binomial theorem.
Context of this
Lessons
Today’s lesson will build off of terminology and ideas from last class.
To Do Today!!!
a. Task 1: A look back at some concepts
i. Rational Functions and Transformations with Vector Notation
ii. Exponential and Log Solving
iii. Quadratic Solving
b. Task 2: A look at Geometric Sequences and Sigma notation/Series
i. Geometric Sequences
ii. Sigma Notation
c. Task 3: Some Practice Problems
B. Fast Ten (Skill Focus)
1. Given f(x)…
a. Write the equations of the asymptotes.
b. State the domain and range of f.
c. Write the equation of f(x)
d. Its image y = g (x) after a translation
6
4
with vector  .
e. Write the equation of g(x).

f.
State the domain and range of g(x).
g. Draw a sketch of g(x) in a different color on the axes provided.
IB SL 1 - Lesson 5.2: Geometric Sequences, Sigma Unit 5 – Series and Sequences
Notation, and Series
2.
Mr. Smith invested 2500 Euro at an annual rate of interest of 3.5 percent,
compounded daily.
(a) Find the value of Smith’s investment after 12 years. Give your answer to the
nearest Euro.
(b) How many complete years will it take for Smith’s initial investment to
quadruple in value?
(c) What should the interest rate be if Smith’s initial investment were to double in
value in 10 years?
3. The equation x2 + 2kx + 3 = 0 has two equal real roots. Find the possible values of k.
IB SL 1 - Lesson 5.2: Geometric Sequences, Sigma Unit 5 – Series and Sequences
Notation, and Series
Geometric Sequences (HUGE TOPIC # 2)
Arithmetic Sequences
8, 11, 14, 17, …
6, 2, -2, -6, -10, …
-15, -8, -1, 6, 13, …
14.4, 18.05, 21.7, 25.35, 29, 32.65, …
Geometric Sequences
100, 50, 25, 12.5,…
2, 4, 8, 16, 32, 64, …
1/3, 1/6, 1/12, 1/24, …
25, 43,75, 76.563, 133.98, …
With your understanding of sequences, try and come up with a definition of what an
Geometric Sequence is
In an Geomotric Sequence…
This is the “recursive formula” for any Geometric sequence!!!
This is the formula to find the nth term of any arithmetic
sequence. See if you and your table can figure out
what it is saying and try and decipher how it works. Use
concrete examples to get you started.
The meaning of this formula…
un+1 =
Explanations for Arithmetic Sequences
un = u1 (rn-1)
IB SL 1 - Lesson 5.2: Geometric Sequences, Sigma Unit 5 – Series and Sequences
Notation, and Series
In an arithmetic sequence, each term can be obtained by multiplying the previous term by
a constant value. This value is called the Common Ratio… or r.
For any arithmetic sequence…
The nth term for an arithmetic sequence
un+1 = un(r)
un = u1 (rn-1)
We can now find any term of an arithmetic sequence by adding common differences to the
pervious term. If we follow the below logic we will arrive at the desired formula…
u1  First Term
u2  u1  r
u3  u2  r  (u1  r) r  u1  r 2
u4  u3  r  (u1  r 2 ) r  u1  r 3
u5  u4  r  (u1  r 3 ) r  u1  r 4
...
un  u1  (r n 1 )
Examples:
a. Find the 9th term of the geometric
sequence 1, 4, 16, 64…
b. In a geometric sequence u1 = 864
and u4 = 256. Find the common ratio
un  u1  r n 1
un  u1  r n 1
u1  1 and r  4
u4  u1  r 4 1  u1  r 3
u9  1 4 91  1 4 8
256  864 r 3
256
r3 
864
u9  65536
256 3 2
256
3
r
or     r
864  3
864
1

IB SL 1 - Lesson 5.2: Geometric Sequences, Sigma Unit 5 – Series and Sequences
Notation, and Series
Sigma () Notation and Series (HUGE TOPIC # 3)
This section mainly has to do with adding the terms of a sequence. Adding the terms of a
sequence gives a series.
A Sequence
A Series
u1  u2  u3  u4  ... un
u1, u2 , u3, u4 , ..., un
Sigma () Notation
n
 ui

i1
This means… the sum of the first n terms of a sequence. You read this…
“The sum of all the terms ui from i=1 to i=n.
Example 1: The arithmetic sequence 8, 14, 20, … has first term 8 and common difference 6. A
general rule for the nth term of this sequence is un = 6n + 2.
What then… does this mean given the sequence un above.
5
(6n  2)
n 1
Please complete your work here!
IB SL 1 - Lesson 5.2: Geometric Sequences, Sigma Unit 5 – Series and Sequences
Notation, and Series
Example 2:
4
a. Write the expression
 (x 2  3) as a sum of terms.
x 1
b. Calculate the sum of these terms.
4
 (x 2  3)
x 1
 (12  3)  (2 2 
3)  (32  3)  (4 2  3)
4
 a.  (x 2  3)  2 1 6 13
x 1
 b.  2 1 6 13  18
Task 3: Practice Problems
Pg. 168 6D: 1
Pg. 169 6E: 1 – 6 All Problems.
Pg. 171 6F: 1-3 All Problems.
Please get this work checked off before you leave today!