Tag Archives: difference

geomseq

Geometric Sequences

I’ve added a script which helps to understand geometric sequences.

Suppose you were to draw an equilateral triangle on a sheet of paper. It might look something like this:

Now suppose that you draw lines connecting the midpoints of each of the edges of this triangle. This will dissect the larger triangle into four smaller triangles, each of which are equilateral. Three of these smaller triangles will be oriented in the same direction as the original triangle, whereas one will not. Consider the second image below, with the three triangles with the same orientation as the original triangle numbered.

We can continue to draw lines connecting the midpoints of the edges of the marked triangles and counting the resulting triangles that have the same orientation as the original triangle and we see that a pattern emerges.

What one notices is that each time we draw a new triangle by connecting the midpoints of the marked edges, we wind up with three times the number of triangles that were in the previous picture. So (assuming we had enough space) we could draw out the figure that would be the result of doing any number of these dissections. However, if we are only interested in knowing the number of triangles that each image will contain, we can take advantage of the fact that this pattern represents a geometric sequence.

A geometric sequence is a sequence with an initial term, a1 and a common ration, r, where each term after the initial term is obtained by multiplying the previous term by the ratio (a1 cannot be zero, and r cannot be zero or one).

In a geometric sequence, if we know the first term and the ratio, we can determine the nth term by the formula

an = a1*rn – 1

Similarly, if we know the first term and the ratio, we can determine the sum of the first n terms in a geometric sequence by the formula:

Sn =
a1(1 – rn)
1 – r

For the previous example with the triangles pointed in the same direction, we can show the results in the following table:

Drawing Number Number of Triangles ratio sum number sum value
a1 1 3 S1 1
a2 3 3 S2 4
a3 9 3 S3 13
a4 27 3 S4 40
a5 81 3 S5 121

The script is available at http://www.learninglover.com/examples.php?id=34

Other Blogs that have covered this topic:
Study Math Online

arithseq

Arithmetic Sequences

Arithmetic Sequences

I’ve added a script which helps to understand arithmetic sequences.

At a previous job of mine, there was a policy of holding a dinner party for the company each time we hired a new employee. At these dinners, each employee was treated to a $20 dinner at the expense of the company. There was also a manager responsible for keeping track of the costs of these dinners.

In computing the costs, the manager noticed that each time there is a new dinner, it was $20 more expensive than the last one. So if we let a1 represent the cost of the first dinner, and let ai represent the cost of the ith dinner, then we see that ai = ai-1 + 20. Sequences like this, where t arise quite often in practice and are called arithmetic sequences. An arithmetic sequence is a list of numbers where the difference between any two consecutive numbers is constant.

For the example above, the term an will represent the cost of dinner after the nth employee has joined the company (assuming that no employees have left the company over this time period). Also the term Sn will represent the total cost the company has paid towards these dinners.

Before we continue with this example, consider the following table which lists the first five terms of an arithmetic sequence as well as the common difference and the first five sums of this sequence.

term number term value diff sum number sum value
a1 4 3 S1 4
a2 7 3 S2 11
a3 10 3 S3 21
a4 13 3 S4 34
a5 16 3 S5 50

One of the beauties of arithmetic sequences is that if we know the first term (a1) and the common difference (d), then we can easily calculate the terms an and Sn for any n with the following formulas:

an = a1 + d*(n – 1), where d is the common difference.
Sn = n*(a1 + an)/2

We can use these formulas to derive more information about the sequence. For example, if my manager wanted to estimate the cost of dinners once we had added 30 new employees, this would be term a30 of the sequence, which we can evaluate with the above formula by a30 = a1 + d*(n – 1) = 0 + 20*(30 – 1) = 0 + 20 * 29 = 580.

The script is available at http://www.learninglover.com/examples.php?id=33.

Other Blogs that have covered this topic:
Study Math Online

Fraction Arithmetic

Fraction Arithmetic

I hope everyone had a good holiday season. I certainly enjoyed mine. Over this season, I had a chance to speak with some youth and their parents. Funny that whenever we discuss that I have a PhD in applied mathematics, the topics of the children struggling in mathematics and the possibility of tutoring their children always seem to come up. I have no problem with tutoring and I actively participate in such sessions in my spare time. However I will say that it is sometimes a difficult task to do this job over such a short time period. Needless to say, I felt bad that I couldn’t have been of more assistance.

So, this being the holiday season and all, I decided to make somewhat of a new years resolution to focus this site more towards some of the things that the youth struggle with to hopefully be able to answer some of their questions.

With that being said, the first area that I decided to look at was fractions. This is one of the first areas where the youth begin to dislike mathematics. I feel like regardless of how much teachers and professors speak of the importance of understanding these processes, many students simply never grasp the procedures involved, partially because they never get used to the rules associated with these matters.

In this first script on fractions, I’ve focused on four types of problems corresponding to the four basic operations of arithmetic: Addition, Subtraction, Multiplication and Division.

To add two fractions of the form

num1
den1
+
num2
den2

We use the formula

num1
den1
+
num2
den2
=
num1
den1
+
num2
den2
=
num1*den2 + num2*den1
den1*den2

Lets take a moment to consider where this formula comes from. In order to be able to add fractions we first need to obtain a common denominator for the two fractions. One way that always works to obtain a common denominator is to multiply the denominators of the two fractions. So in the formula above, the denominator on the right hand side of the equals sign is the product of the two denominators on the left hand side. Once we have a common denominator, we need to rewrite each of the two fractions in terms of this common denominator.

num1
den1
+
num2
den2
=
num1*den2
den1*den2
+
num2*den1
den1*den2

The formula for subtracting fractions is similar, with the notable difference of a subtraction in the place of addition.

num1
den1
-
num2
den2
=
num1*den2 – num2*den1
den1*den2

To multiply two fractions (also known as taking the product of two fractions, the resulting numerator is the product of the two initial numerators, and likewise the resulting denominator is the product of the two initial denominators.

num1
den1
*
num2
den2
=
num1*num2
den1*den2

Finally, remembering that division is the inverse of multiplication, we can derive the formula to divide two fractions by multiplying by the inverse of the fractions:

num1
den1
÷
num2
den2
=
num1
den1
*
den2
num2
=
num1*den2
den1*num2

The next step in each of these operations is to reduce the fraction to lowest terms. One way of doing this is by considering Euclid’s GCD algorithm which is available here.

The script is available to practice your work on fractions at

http://www.learninglover.com/examples.php?id=31

Learning Math through Set Theory

In grade school, we’re taught that math is about numbers. When we get to college (the ones of us who are still interested in math), we’re taught that mathematics is about sets, operations on sets and properties of those sets.

Understanding Set Theory is fundamental to understanding advanced mathematics. Iv wrote these scripts so that users could begin to play with the different set operations that are taught in a basic set theory course. Here, the sets are limited to positive integers and we’re only looking at a few operations, in particular the union, intersection, difference, symmetric difference, and cross product of two sets. I will explain what each of these is below.

The union of the sets S1 and S2 is the set S1 [union] S2, which contains the elements that are in S1 or S2 (or in both).
Note: S1 [union] S2 is the same as S2 [union] S1.

The intersection of the sets S1 and S2 is the set S1 [intersect] S2, which contains the elements that are in BOTH S1 and S2.
Note: S1 [intersect] S2 is the same as S2 [intersect] S1.

The difference between the sets S1 and S2 is the set S1 / S2, which contains the elements that are in S1 and not in S2.
. Note. S1 / S2 IS NOT the same as S2 / S1.
Note. S1 / S2 is the same as S1 [intersect] [not]S2.

The symmetric difference between the sets S1 and S2 is the set S1 [symm diff] S2, which contains the elements that are in S1 and not in S2, or the elements that are in S2 and not in S1.
Note. S1 [symm diff] S2 is the same as S2 [symm diff] S1.
Note. S1 [symm diff] S2 is the same as (S1 [intersect] [not] S2) [union] (S2 [intersect] [not] S1).

The cartesian product of the two sets S1 and S2 is the set of all ordered pairs (a, b), where a [in] S1 and b [in] S2.