Name ________________________
Worksheet 1.5
Sequences and series and Inductive reasoning.
We are currently looking at Logic and syllogisms which are examples of deductive reasoning.
“If this is true then that is true”, in particular if something is true for all cases then it is true for
each specific case.
All birds have feathers, an ostrich is a bird, therefore an ostrich has feathers.
When we reason inductively we find something that is true as a pattern for a small number of
cases and justify extending the concept to all cases.
“The sum of the first n odd numbers is equal to the nth square” is a statement which we can
prove inductively.
We observe that:
1 = 12
1+3 = 22
1+3+5 = 32
Which seems to add weight to our hypothesis.
The induction consists of showing that if the (n+1)th odd number is added to the nth square
then the result is the (n+1)th square.
We are not assuming that what we are trying to prove is true, we are saying IF it is true AND
this formula works then the next case is true too.
i.e. n 2  (2n  1)   n  1 which we know algebraically to be valid. But we do require the first
2
steps in this pattern to be valid for all to be valid. 1+3 =(1+1)2, which is true. ( and 22+5 =32)
This gives the valid foundation on which to build and so our statement is true.
We can also perform the induction visually.
Draw the next two arrays and use the picture to explain why the (n+1)th odd number is 2n+1.
The formal argument for the induction is
n
The statement for case n P(n):
 2k  1  n
2
k 1
n 1
n
(n)   2k  1  2k  1  (2n  1)  n 2  2n  1  ( n  1) 2
k 1
k 1
n 1
i.e.(n)   2k  1 (n  1) 2
k 1
Which is P(n+1) so P(n)  P(n+1) but since P(1) is true this means P(2) is true etc
So all are true and P(n) is true.
Follow a similar process with a formula, a picture and then an induction to prove that:
1. The sum of the first n integers is equal to the nth triangle number.
2. A plane divided by n lines all intersecting each other can always be colored using exactly
two colors.
n
1
1
3.  k  1  n
2
k 1 2
NOTES:
1. The sum of the integers and the sum of the odd numbers are both examples of arithmetic
series.
2. A geometric series however is not defined from one term to the next by addition, but
rather by multiplication.
Eg: The sequence of powers of two is geometric as is the series
1 1 1
   ....
2 4 8
Sequences:
In the world of mathematics, we desire to be witnesses and creators of patterns and
order. Sequences are one way of representing numbers so that we can see patterns and
establish order.
How would you define a sequence?
Provide an example of a sequence:
Let’s strive to arrive at a formal definition of sequence; one that can be universally
accepted among all those who work and play in the wonderful world of mathematics.
Definition of sequence:
Definition of term:
Question to ponder: How do we find terms?
Ex1: Given
an 
n 1 , find the first five terms and the 99th term.
n 1
Ex2: Given the following sequence, write an explicit formula (like the one above) to
identify the nth term.
1, -3, 9, -27, 81, …
Explicitly vs. Recursively Defined Formulas for Sequences:
After studying rabbit populations and observing how fast rabbits could breed under
ideal circumstances, a famous Italian (to be named later) posed the following question:
Question: How many pairs of rabbits will there be in one year under the following ideal
circumstances?
Conditions: 1) Rabbit population begins with one female and one male
2) Rabbits are able to mate at the age of one month, so that at the
end of its second month a female can produce another pair of
rabbits
3) Rabbits will never die (during this year)
4) Females will always produce one pair (from end of second
month on)
Try drawing some bunnies, to reflect this growth…
How many rabbits will there be after the 4th month is complete (end of 4th month)?
How many rabbits will there be in one year (end of 12th month)?
Can you derive a recursively defined formula—where the nth term is defined using the
previous term(s)—to help find how many rabbits there will be, given earlier rabbit
numbers (from prior months)?
Find a formula for the number of diagonals of an n sided polygon.
A partial sum is the sum of a finite number of terms of a sequence. We denote the nth
partial sum as S n . For example, the third partial sum, S 3 , is the sum of the first three
terms. Now that you have been introduced to the idea of a partial sum, make a valiant
attempt at arriving at a solution for the following:
Ex3: Find the first four partial sums and the nth partial sum of the sequence given by
1
an  n
2
Ex4: Find the first four partial sums and the nth partial sum of the sequence given by
an 
1
1

n 1 n  2
Summation Notation:
We mathematicians do not like to write too much. Let’s face it; we are not lazy, we just
want to be efficient in what we write and how we communicate. We have our own
language; constructed using symbols and letters widely accepted in the mathematics
community.
Let’s interpret and evaluate the following:
4
 (k 1)2
k 1
Properties of Sums:
1) Addition/Subtraction Property
2) Constant Multiple Property
Practice Exercises:
1) Find the first seven terms of the given recursively defined sequence:
an  an1  an2  an3
a1  a2  a3  1
2) Find the first four partial sums and the nth partial sum of the sequence an .
an  n  n 1
3) Write the sum using sigma notation in TWO distinct ways.
2  5x  8x2 11x3 14x4  ...  41x13
4) Find both a recursive and an explicit formula for the nth term of the following
sequence:
2,
2 2,
2 2 2,
2 2 2 2 ,...
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 
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:
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 a 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 
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:
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