the reason we go away
Sequences and Series Project Patterns occur everywhere in life especially in mathematics. A pattern can be defined as any sequence of numbers that may be modeled by a mathematical function. A sequence is an ordered list of numbers such as 1, 2, 3, 4. A pattern can be found in a sequence, but a sequence doesn’t always necessarily have a pattern. For some patterns, you can even find a rule that fits them. There are two types of rules: recursive and explicit, and both rules can be used to find the next term in a pattern. The recursive rule has two parts: an initial condition (the value of the first term) and a recursive formula which relates each term after the first term to the one before it. In other words, the recursive rule is used to show how you get from one place in the sequence to the very next term. However, the explicit rule is a rule that can be solved without needing the previous term, but instead using the position of the missing term in the explicit formula. The explicit rule is used to find any missing term in a sequence by using its place in the sequence and doing something to it to find the missing term. Therefore, although both rules can be used in finding missing terms, explicit will allow an easier time finding nonconsecutive missing terms. There are also two types of sequences that recursive and explicit rules can be applied to: arithmetic and geometric. An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This difference is the common difference which the variable d is commonly used to represent it. If a sequence is arithmetic meaning it has a common difference, then a specific rule can be easily found using either the arithmetic recursive formula or the arithmetic explicit formula. The recursive formula is an = an-1 + d, for n>1 and a1 = a. The explicit formula is an = a + (n – 1)d, for n > 1. Using these formulas, you can get specific rules for arithmetic sequences such as 1, 2,