Thursday, June 4, 2009

june/4/09

Arithmetic and Geometric series.

We started the class with a quick probability question, one that was brought up by a student.

When solving questions like these, always try to go back to the basics. Probability is the number of favorable outcomes over the possible outcomes. From that, we have something to work with.
Finding the possible outcomes, or Sample Space.
• The question states that "4 men and 4 women" would be chosen. This means that out of the 7 men, 4 will be chosen (7C4), and out of 10 women, 4 will be choose (10C4).
Finding the favorable outcomes.
• Since the question said that, "Allen and Bridget will be among these 8 chosen people", we already know that they are part of the possible outcomes. They are represented by the green and blue "1"'s.
• Out of the 7 men, only one has been chosen, (Allen). This leaves us with 6 more men, and 3 more spots for men, 6C3.
• Out of the 10 women, only Bridget has been chosen. This leaves us with 9 more women, and 3 more spots for women, 9C3.
As a result, you would get what is shown on the image above. (:

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Today's class was focused on:
1. Arithmetic and geometric sequences
2. Series
3. Arithmetic and geometric series.
4. Sigma Notation
Arithmetic sequences
An arithmetic sequence is a sequence (ordered list of numbers), where a fixed number(common difference) is found between two consecutive terms. (negative numbers are added too! )
This means each term is going up or down by the same number.
When wanting to find the nth term in an arithmetic sequence, refer to the equation below. (use Carl Friedrich Gauss's 7year old story to help you remember the equation.)

Geometric sequences
Geometric sequences are like arithmetic sequence, but instead of adding its multiplying. This means instead of a common difference, there's a common ratio.
When finding the nth term in a geometric sequence, refer to...
Series
Series is defined as, the sum of terms in a sequence. (Sn, where S reads as "sum of" and n would be the rank of the nth term. ex. S4 = sum of the first 4 terms.)

If we were to have a sequence of 1,2,3,4,5,6,7 etc, the series would consist of 1,3,6,10,15. Why? Well the first term is a given, 1. The second term would be 3, because that was the sum of the first and second term from the sequence, (1+2). The third term is the sum of the first, second, and third terms from the sequence. (1+2+3). The orange circled numbers in the above picture are the ranks.

Arithmetic series
Arithmetic series is the sum of numbers in an arithmetic sequence. This would be helpful if you are asked to find the "sum of integers from 1 to 5000" for example. Where n would be 5000, because there are 5000 terms, a would be 1, because that is the value of the first term, and d would be 1, because that's the common difference. (numbers are going up by one)
Or you might be asked "What is the sum of all multiples of 7 between 1 & 5000".
What you know:
• From the sequence of multiples of 7 between 1-5000, first term is 1.
• Last term is 4998
• common difference is 7.
What we need to know:
• number of terms within that sequence.
By using the arithmetic equation, tn = a + (n-1)d, plug in known values, do some grunt work, and you'll be able to find n. (or you can just take the last number, 4998, and divide by 7, to see how many times 7 can go into it.4998/7 = 714) Once you've found the number of terms, plug it into the arithmetic series equation. ta-da~
Now don't get too carried away with questions a and b. These kind of questions wont be asked on the exam, but you'll need to know the methods of a&b in order to solve c. (c = a question likely asked on the exam).

Logic:( The sum of all integers 1-5000) - (sum of all multiples of 7) = sum of all integers not multiples of 7. This "build up" to a question, is called scaffolding.

Geometric series.
Sum of numbers in a geometric sequence.

Sigma Notation
Is the shorthand way of writing a series, also known as the weird looking "E". Sigma is really confusing, if you don't know how to read it. The n=1 tells you the value of the first term, which is 1. The 4 on top of the sigma is nth number of term to stop at. The (2n-3) is the "rule" or equation you follow.

bye guys! good luck on the exams and your DEVs!
mary
The next scribe is jonno!