![]() Find the recursive and closed formula for the sequences below. Then we have, Recursive definition: an ran 1 with a0 a. Suppose the initial term a0 is a and the common ratio is r. Then we will investigate different sequences and figure out if they are Arithmetic or Geometric, by either subtracting or dividing adjacent terms, and also learn how to write each of these sequences as a Recursive Formula.Īnd lastly, we will look at the famous Fibonacci Sequence, as it is one of the most classic examples of a Recursive Formula. A sequence is called geometric if the ratio between successive terms is constant. I like how Purple Math so eloquently puts it: if you subtract (i.e., find the difference) of two successive terms, you’ll always get a common value, and if you divide (i.e., take the ratio) of two successive terms, you’ll always get a common value. Then, we either subtract or divide these two adjacent terms and viola we have our common difference or common ratio.Īnd it’s this very process that gives us the names “difference” and “ratio”. For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, Sa11r, where a1 is the first term and r. And adjacent terms, or successive terms, are just two terms in the sequence that come one right after the other. The geometric sequence is sometimes called the geometric progression or GP, for short. Well, all we have to do is look at two adjacent terms. A geometric sequence is a sequence of terms (or numbers) where all ratios of every two consecutive terms give the same value (which is called the common ratio). It’s going to be very important for us to be able to find the Common Difference and/or the Common Ratio. Comparing Arithmetic and Geometric Sequences
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