![]() How many common differences are needed to get from the first to the third term? If you know the first term of a sequence ( ), how many common differences do you need to add to it to get to the second term of the sequence ( )? Since you seek the very next term, only one difference is needed: Generalizing this linear equation approach leads to a description that applies to any Arithmetic Sequence: The last equation above uses this linear model, and provides the fastest way to calculate the Nth term of the sequence. Note that if a sequence starts with a 5 then increases by 3 from one term to the next, this situation can be modeled using a linear equation with 5 as its y-intercept and 3 as its slope (with the domain restriction that “n” must be a positive integer). This is a “Closed Form” definition (you only need to know the term number) Specify a rule (based on the term number) for calculating the “Nth” term This is a “Recursive” definition (you must know the previous term) Specify the first term, with a rule to get you from each term to the next Specify the first term and the common difference Specifying the first three or four terms is enough to demonstrate the common difference These insights allow a complete description of an Arithmetic Sequence to take a number of forms: A more intuitive way of reading this equation is “Any term may be calculated by adding the common difference to the previous term”. If refers to the “Nth” term, then has a subscript that is one less than N, and therefore refers to the term that immediately precedes. #Sum of arithmetic sequence plusThis would be read as “A sub N is equal to A sub N-1 plus the common difference d”. Since every line above follows the same pattern, the whole process can be described a bit more generally and compactly by using a variable as a subscript: The existence of a common difference ( ) allows us to calculate terms in a generic way: Algebraic Description Of An Arithmetic Sequence No matter what value it has, it will be the difference between all consecutive terms in that Arithmetic Sequence. It can be a whole number, a fraction, or even an irrational number. The common difference can be positive or negative. It can be calculated by subtracting the previous term from any term: or, etc.Įvery Arithmetic Sequence has a common difference between consecutive terms. If you add this value to any term, you end up with the value of the next term. Since all of the terms in an Arithmetic Sequence must be the same distance apart by definition (3 apart in the example above), the magnitude of this distance is given a formal name (the common difference) and is often referred to using the variable (for Difference). So represents the value of the first term of the sequence (5 in the example above), and represents the value of the seventh term of the sequence (23 in the example above). The one is a “subscript” (value written slightly below the line of text), and indicates the position of the term within the sequence. This notation is read as “A sub one” and means: the 1st value of the sequence represented by “a”. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation Where n is the number of terms in the sequence, a 1 is the first term in the sequence, and a n is the n th term, and d is the constant difference between each term.In the example above 5 is the first term, or starting term, of the sequence. The sum of a finite arithmetic sequence can be found using the following formula, #Sum of arithmetic sequence seriesFor example, 2 + 5 + 8 = 15 is an arithmetic series of the first three terms in the sequence above. Arithmetic sequence vs arithmetic seriesĪn arithmetic series is the sum of a finite part of an arithmetic sequence. This formula allows us to determine the nth term of any arithmetic sequence. Therefore, the 100th term of this sequence is: Using the above sequence, the formula becomes: Where a n is the n th term, a 1 is the initial term, and d is the constant difference between each term. Fortunately, the nth term of an arithmetic sequence can be determined using This is simple for the first few terms, but using this method to determine terms further along in the sequence gets tedious very quickly. To expand the above arithmetic sequence, starting at the first term, 2, add 3 to determine each consecutive term. For example, the difference between each term in the following sequence is 3: Home / algebra / sequence / arithmetic sequence Arithmetic sequenceĪn arithmetic sequence is a type of sequence in which the difference between each consecutive term in the sequence is constant. ![]()
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