DISCRETE FUNCTIONS: Series and Sequences
This investigation is a guided approach to the discovery and understanding of series and
sequences. You have already worked with many different types of sequences (but were never
really told at that time that they were sequences). This guide will also serve as your notes for
the introduction of discrete function. Be sure to analyze and answer these questions thoroughly.
You can also use the textbook (Sections 6.1, 6.2, 6.4 – 6.7) to help you out by looking closely
at their examples, definitions and especially the “Key Concepts” sections.
PART I. Answer the following questions, keeping in mind that you are also watching for special
characteristics, patterns, relationships, formulas, etc. that will help you understand sequences.
1. The following is a chart representing different size equilateral triangles.
Side Length
1 cm
2 cm
3 cm
4 cm
5 cm
Perimeter
….
20 cm
….
….
….
What is the relationship between perimeter and side length of an equilateral triangle?
2. The cells in a special culture divide and double every hour.
a) If a culture starts with 500 cells, how many cells will there be after 1 hour?
b) Create a table showing the number of cells at the end of each hour for the first 5 hours?
c) What type of relationship exists between the number of cells and the time?
d) Create an equation that relates the number of cells to the time.
n cm
3. In each of the next questions a pattern can be found. Find the pattern and write the next three numbers.
(these are called the next three terms)
a) 1, 3, 5, 7, _____, _____, _____
c)
b) 12, 24, 48, 96, _____, _____, _____
3 5 7
, , , _____, _____, _____
4 9 16
d) -1, 3, -9, 27, _____, _____, _____
The above are all examples of SEQUENCES.
A sequence is a set of numbers, shapes, letters, etc. that are in a distinct or recognizable pattern.
…………………………………………………………………………………………………………...
PART II. In the following few questions pay particular attention to the special definitions and notations that
will be used for sequences.
1. In the sequence 3, 6, 9, 12, 15, 18, . . .
12 is referred to as the fourth term
and we also use a special notation t4 = 12 to represent the fourth term
Find: t2 = _____ t6 = _____ (you should notice that these correspond to Part I #1)
[Also, the “formula” that you found for the perimeter should have been 3n]
Therefore, in general tn = 3n is called the general term for the sequence
Find: t30 = _____
t50 = ______
t100 = _____
What does t100 mean?
What does tn mean?
How is the notation tn similar to the notation f (n) from earlier in the course?
2. From the given general term, find the first 4 terms of each of the following sequences.
(a couple of them have already been done for you)
a) tn = 2n + 5
t1 =
2(1) + 5 = 7 is the first term
b) tn = 3n
t1 = 31 = 3 is the first term
t2 =
t2 =
t3 =
t3 =
t4 =
t4 =
c) tn = -3n + 1
d) tn = 3(2)n – 1
t1 =
t1 =
t2 =
t2 =
t3 =
t3 =
t4 =
t4 =
3. Let’s look at the sequence 7, 10, 13, 16, 19, 22, . . . a little differently (you fill in the blanks)
t1 = 7 . . . . . . . . . . . . = 7 + (0)(3)
t2 = 10 = 7 + 3 . . . . . . .
= 7 + (1)(3)
t3 = 13 = 7 + 3 + 3 . . . . . . = 7 + (2)(3)
t4 = 16 = 7 + 3 + 3 + 3 . . . . . = 7 + (3)(3)
t5 = 19 = 7 + 3 + 3 + 3 + 3 . . . = 7 + (__)(3)
t6 = 22 = 7 + 3 + 3 + 3 + 3 + 3 . . = 7 + (__)(3)
so
t20 = ……………………………… = 7 + (__)(3) = ________
t50 = ……………………………… = 7 + (__)(3) = ________
and in general tn = ___________________ (this is also referred to as nth term of the sequence)
This is also an example of an ARITHMETIC SEQUENCE [#2 a) and 2 c) are also arithmetic]
In your own words, what do you think an arithmetic sequence is?
List as many properties as you can.
In general every arithmetic sequence has:
- a first term which we will let be a
- a common “difference” which we will let be d
(and this d = t2 – t1 = t3 – t2 = t4 – t3 = etc)
With this in mind and using results from Part II #3 above complete the following:
t1 = a
t2 = a + d
t3 = a + 2d
t4 = a + 3d
t5 = a + 4d
t6 =
t10 =
t50 =
tn =
(this is referred to as general term of any arithmetic sequence)
PART III. Let’s look at the sequence 3, 3(2)1, 3(2)2, 3(2)3, . . .
Find:
t5 =
t10 =
t25 =
tn =
This is an example of a GEOMETRIC SEQUENCE.
In your own words, what do you think a geometric sequence is? List as many properties as you can.
In general every geometric sequence has:
- a first term which we will let be a
- a common “ratio” which we will let be r
(and this r =
t2
t
t
= 3 = 4 = etc.)
t1
t2
t3
With this in mind and using results from prior parts of this assignment complete the following:
t1 = a
t2 = ar
t3 = ar2
t4 = ar3
t5 = ar4
t6 =
t7 =
t10 =
t50 =
tn =
HOMEWORK:
p. 433 #2,3,4,7,11
p. 441 #3,4,6,7,8,16,20,23
p. 452 #1,2,3,4,5,6,7,9,13
(this is referred to as general term of any geometric sequence)
PART IV. Recursion Formulas
 With an explicit formula, you can calculate the nth term of a sequence without knowing any previous terms
 With a recursion formula, you can calculate the nth term of a sequence only when you know one or more
previous terms.
 A recursion formula has two parts: 1) The value(s) of the first term(s).
2) A (recursion) equation that shows how to calculate each term
from the term(s) before it.
EXAMPLE 1: Find the first five terms of the sequence.
a) t1  5; tn  2tn 1  3
b) t1  2; tn  tn 1  3
c) t1  1; t2  2; tn  4tn 2  3tn 1
EXAMPLE 2:
Write an explicit formula for the sequence
determined by the following recursion formula.
EXAMPLE 3:
Write a recursion formula for the sequence
determined by the following explicit formula.
a) t1  2; tn  tn 1  3
a) tn  4n
PART V. Series
Arithmetic Series
 An arithmetic series is the sum (total) of the values of the terms of an arithmetic sequence.
i.e., Sn  t1  t2  t3  t4  ...  tn
 If the first two terms of an arithmetic series are known, the sum of the first n terms can be found using the
n
following formula: Sn   2a  (n  1)d  where a is the first term
2
d is the difference between any two terms
 If the first and last terms of an arithmetic series are known, the sum of the first n terms can be found using the
n
following formula: Sn   a  tn 
where a is the first term
2
tn is the last term
EXAMPLES: Find the sum of each arithmetic series.
a) 5  7  9 
b) 4  1  6 
 27
d) 34  30  36 
2
 91
e) a  1; t15  13
c) S20 for 1 + 3 + 5 + 
f) t1  2.4; t18  5.8
Geometric Series
 A geometric series is the sum (total) of the terms of a geometric sequence.
 The sum of the first n terms of a geometric series can be found using the following formula:
Sn 
a  r n  1
r 1
where a is the first term
r is the ratio between any two terms
EXAMPLES: Find the sum of each geometric series.
a) S10 for 7  21  63 
b) a  2, r  2, n  9
d) 3  6  12   384
1
2
e) 2  1   
1
64
c) f (1)  16, f (2)  8, n  9
f) 16  8  4  
1
8
PART VI. Pascal’s Triangle
1. Complete Pascal’s triangle below:
2. What patterns do you notice in Pascal’s Triangle?
3. Expand each of the following binomials.
(a + b)0 =
(a + b)1 =
(a + b)2 =
(a + b)3 =
(a + b)4 =
4. Blaise Pascal, French Mathematician discovered a pattern in the expansion of (a + b)n.
What patterns do you notice?
5. Expand each binomial using Pascal’s Triangle
a) (a + b)6
b) (2x – 3)5
b) (2m + 3n2)5
d) ( 2 − 𝑦 2 )
6. How many terms will there be if you expand (x + 2y)20?
7. What is the second term in the expansion of (x + 6)7
8. What is the 5th term in the expansion of (3y – 4)8?
9. What is the coefficient of x3 in the expansion of (x + 6)6?
10. What is the coefficient of y4x2 in the expansion of (y + 3x)6?
𝑦
4