Download Notes on Arithmetic Series Part I

Notes on Arithmetic Sequence
Name_________________________
An arithmetic sequence is a series of numbers in which the difference between each of the
terms is constant (i.e., it is always the same). For example, examine the sequence of numbers
below:
3, 7, 11, 15, 19, ...
In this series, the difference, or, the quantity added to one term in order to get the next term, is
constant. It is 4. Once again, it is important for the sake of consistency to refer to the difference
as the quantity added (not subtracted) to one term in order to get the next term. It is possible for
the difference to be a negative quantity.
Is the sequence an arithmetic sequence? If so, what is the difference?
1.
9, 17, 25, 33, 41
2.
1, 2, 4, 7, 11, 16
3.
12, 7, 2, -3, -8,...
4.
1.4, 1.6, 1.8, 2.0,...
5.
4, 6, 9, 11, 14, 16, 19,...
6.
1 5 11 7
, , 2, ,
2 4
4 2
--------------------------------------------------------------------------------------------------------------------If you recall from the study of sequences during Math 1, the closed form definition is a formula
which allows one to find the value of the nth term of a sequence. The closed form definition will
serve the same purpose in this study of arithmetic sequence. In other words, if one wants the
value of the 20th term of an arithmetic sequence, then he/she should be able use the closed form
definition to obtain it.
Activity to find the closed form definition
Consider the arithmetic sequence below:
2, 5, 8, 11, ...
Let's place this sequence in a two-column table.
Term Number (n)
Term Value ( Tn )
1
2
2
5
3
8
4
11
On the back of this page, graph this arithmetic sequence on a two-dimensional coordinate plane,
letting x represent the term number and y represent the term value.
Why do you think the graph looks like this? (Hint: think about slope)______________________
_____________________________________________________________________________
The equation of a line in standard form is y
-
=mx+b
y is Tn
x is n
m is the slope or the difference (d) between 2 consecutive numbers in the sequence
b represents the y-intercept (when x  0 ). It is (the term previous to the 1st term or T1)
Let's substitute those letters into the equation of a line. Here is the final result:
This is the closed form definition for any arithmetic sequence!!! It can be quickly used to find
the value of any term in the sequence.
How would you find the 0th term of an arithmetic sequence? _____________________________
--------------------------------------------------------------------------------------------------------------------7.
Consider the following arithmetic sequence: 8, 14, 20, 26, ...
What is the value of the 23rd term?__________
Which term in the sequence has a value of 242?_____________
8.
Consider the following arithmetic sequence: 21, 10, -1, -12, ...
What is the value of the 50th term?___________
Which term in the sequence has a value of -441?____________
--------------------------------------------------------------------------------------------------------------------Calculating the Sum of a Series
Consider the following series: 1, 6, 11, 16, 21,...,111, 116
What is the sum of all of the numbers in this series?______________
Try to think of different methods that could be used to find the sum.
--------------------------------------------------------------------------------------------------------------------Let's consider a much shorter series now: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.
Using any method, find the sum of all of the numbers in this series.______________
What is the sum of the 1st and the last (10th) numbers in the series?____________
What is the sum of the 2nd and the next-to-last (9th) numbers in the series?_____________
What is the sum of the 3rd and the 8th numbers is the series?_____________
In total, how many pairs like these can be made until the entire series is exhausted? __________
How does this number compare to the total number of terms (n) in the series?_______________
So, if one knows that constant sum, and if one knows the number of pairings that can be made,
he/she can quickly obtain the sum of the series. In other words, the sum of the entire series will
just be that constant sum (which can be formed by adding the first and last term in the series)
multiplied by the number of them that there are.
SUM OF THE FIRST N TERMS OF ARITHMETIC SERIES
If you were given the arithmetic series 5, 7, 9, 11, 13, … and were asked to find the sum of the
first 50 terms, what information would you need in order to do so? ________________________
______________________________________________________________________________
Find the sum of the first 50 terms of this arithmetic series. ____________________
Homework on Arithmetic Series
Name_________________________
For Questions 1-4, is the sequence an arithmetic sequence? If so, what is the difference?
1.
2, 4, 8, 16, 32, ...
2.
7, 20, 33, 46, 59, 72
3.
10, 9, 7, 4, 0, -5, ...
4.
-8, -11, -14, -17, ...
5.
Make a finite arithmetic sequence with six terms and a difference of 9.
6.
Consider the following arithmetic series: 5, 9, 13, 17, 21, 25, ...
A)What is the value of the 25th term?
B)Which term has a value of 521?
C)What is the sum of the first 10 terms of the series?
D)What is the sum of the first 100 terms of the series?
7.
Consider the following arithmetic series: 10, 8, 6, ..., -100, -102, -104
A)What is the value of the 34th term?
B)How many terms are in the series?
C)What is the sum of all of the terms in the series?
8.
Consider the following arithmetic series: 1, 2.25, 3.5, 4.75, ...
A)What is the value of the 71st term?
B)What is the value of the first term that is greater than 500?
C)What is the sum of the first 76 terms of the series?
D)The first n terms have a sum of 6162.75. What is the value of n?