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Mathematical Sequences and Pascal’s Triangle

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Overview:  What Are Mathematical Sequences?
A mathematical sequence is a function that follows a specific rule and a specific pattern.  For example, a number set A consists of {3, 5, 7, 9, 11 …}.  Such a number set is easily recognized as the set of odd numbers.  If there is no last term, the sequence is infinite, and if there is a last term, the sequence is finite.

Finding the Rule
In order to find the rule it follows, pair each number with a positive integer first,  and then find the nth term in the sequence.  Continuing with the example, the first term in the series is 3, so pair (1,3).  The second is (2,5); the third is (3,7);  the fourth is (4, 9); the fifth is (5, 11).  The nth term defines the rule: an= 2n +1.  Check to see if the rule is a true expression.  Does the sixth term, a6, follow the rule?  If so, the sixth pair will be (6, [2(6) +1] ) or (6, 13).

Arithmetic Sequences
An arithmetic sequence is a sequence in which the difference between successive terms is a constant, called the common difference or d.  For example, a2 – a1=d and a2 = a1 +d.  Generally, an=a1 +(n-1)d.  This means that if the first term of an arithmetic sequence is 3 and the difference is 2, the 20th term will be  a20 = 3 + (20 -1)2, or 3 +38 = 41.

Geometric Sequences
A geometric sequence is a sequence in which the ratio of successive terms is a constant, called the common ratio.    For example, an +1/an is the same ratio, no matter where the points next to each other are in the sequence.  Using the geometric sequence, {4, 8, 16, 32…}, it can be shown that 8/4 =16/8 =32/16, and so on.  Generally, an=a(rn-1), where a without the subscript is equal to the first term in the sequence and r is the geometric constant.  Therefore, a8 = 4(27) or 4(128), or 512.

Pascal’s Triangle:  An Application of Sequences
One of the most famous sequences in mathematics is known as “Pascal’s Triangle”, named after the mathematician who summarized its properties.  It is a sequence of binomial coefficients, arranged so that the each number in the triangle is the sum of the two that are above it.  The properties of these sequences form the arrangements in probability theory.

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