Appendix: Discrete Algebra
We all start math with counting. Now we are back to the counting again. But this time is called the sequence and series.
A sequence is a function defined on the positive integers or on a subset of consecutive positive integers starting with 1. There are two types of sequences. One is called the arithmetic sequence. One is an arithmetic sequence if and only if an = a1 + (n - 1)d for the sequence with first term a1 and nth term an. One is a geometric sequence if and only if an = a1r n-1 for the sequence with first term a1, and nth term an with r ¹ 0.
A series is the indicated sum of the terms of a sequence. For an arithmetic series in which a1 is the first term, d is the common difference, an is the last term, and Sn is the value of the series: Sn = n(a1 + an ) / 2 and Sn= n[2a1 + (n - 1)d] / 2. For a geometric sequence in which a1 is the first term, an is the last term, r is the common ratio (r ¹ 1), and Sn is the value of the series: Sn = a1 a1rn / 1 r and Sn = a1 - anr / 1 r. The value of an infinite geometric series with first term a I and common ratio r, |r| < 1, is given by the formula: S = a1 / 1 r.
Probability is a very important part of peoples lives. Businesses and scientific researches use probability the most for predicting the revenue growth and experiments outcomes. But ironically, the probability is one area of math that has the least to talk about. If you want to be good at probability, you have to do as much problems as possible in a wide range of probability problems. What you can do now is to understand some basic concepts about probability. The first one is the Fundamental Counting Principle. It states that if one event can occur in m ways and a second event can occur in n ways, the pair of events can occur in mn ways. The most important two concepts about probability are permutation and combination.
A permutation of a number of objects is any arrangement of the objects in a definite order. For all positive integers n and r, where r -- n, the number of permutations of n things taken r at a time is nPr = n! / (n r) ! For each positive integer n, n! = nx(n - 1)x(n - 2)x . . . x 3 x 2 x 1. Also, 0! = 1. There is a special case where permutations in the denominator and numerator repeat, and this is called the Permutations with Repetitions. It states that for all positive integers n and r, where r £ n, the number of distinguishable permutations of n objects, r of which are alike, is: nPn / rPr = n! / r!
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