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Patterns are found everywhere and anywhere in life. There are endless patterns in nature, from the shapes of snowflakes to the atomic structure of every material. Sequences are just one example of these kinds of patterns. Specially, sequences are sets of numbers which people use every day for problem solving and many methods have been developed to create and analyze sequences. Arithmetic SequencesSequences are not just any set of numbers. They follow a rule that can be expressed through equations such as the equation where represents the term in the sequence. This rule creates the sequence 0, 1, 2, 3, etc. This is one of the most basic kinds of sequences and it is one that is used every day. To find the equation of a sequence, first it must be determined to be an arithmetic sequence or a geometric sequence. Arithmetic sequences are sequences that the terms differ by adding or subtracting a certain amount from the previous term. This amount is called a common difference and is found by simply subtracting any term by the previous one. Take this arithmetic sequence as an example. -1, 2, 5, 8, 11, 14, 17, 20The common difference is found through . All of the terms should be the same so for the sequence above . This not only gives us a common difference of 3, but also proves that it is an arithmetic sequence.The next step is to find the equation of the sequence so that any term can be found. The formula for arithmetic sequences isWhere is the first term of the sequence, is the term, and is the common difference. This equation is easily simplified when the common difference is found. For the example sequence it would be To simplify, first distribute the 3 to Then combine like termsNow that we have the equation, any term in the sequence can be found meaning missing parts of the...

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