![]() ![]() For example, to calculate the sum of the first 15 terms of the geometric sequence defined by a n = 3 n + 1, use the formula with a 1 = 9 and r = 3. In other words, the nth partial sum of any geometric sequence can be calculated using the first term and the common ratio. S n − r S n = a 1 − a 1 r n S n ( 1 − r ) = a 1 ( 1 − r n )Īssuming r ≠ 1 dividing both sides by ( 1 − r ) leads us to the formula for the nth partial sum of a geometric sequence The sum of the first n terms of a geometric sequence, given by the formula: S n = a 1 ( 1 − r n ) 1 − r, r ≠ 1. Subtracting these two equations we then obtain, R S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n Multiplying both sides by r we can write, S n = a 1 + a 1 r + a 1 r 2 + … + a 1 r n − 1 Therefore, we next develop a formula that can be used to calculate the sum of the first n terms of any geometric sequence. However, the task of adding a large number of terms is not. For example, the sum of the first 5 terms of the geometric sequence defined by a n = 3 n + 1 follows: is the sum of the terms of a geometric sequence. In fact, any general term that is exponential in n is a geometric sequence.Ī geometric series The sum of the terms of a geometric sequence. In general, given the first term a 1 and the common ratio r of a geometric sequence we can write the following:Ī 2 = r a 1 a 3 = r a 2 = r ( a 1 r ) = a 1 r 2 a 4 = r a 3 = r ( a 1 r 2 ) = a 1 r 3 a 5 = r a 3 = r ( a 1 r 3 ) = a 1 r 4 ⋮įrom this we see that any geometric sequence can be written in terms of its first element, its common ratio, and the index as follows:Ī n = a 1 r n − 1 G e o m e t r i c S e q u e n c e Here a 1 = 9 and the ratio between any two successive terms is 3. For example, the following is a geometric sequence, 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 geometric sequence A sequence of numbers where each successive number is the product of the previous number and some constant r., or geometric progression Used when referring to a geometric sequence., is a sequence of numbers where each successive number is the product of the previous number and some constant r.Ī n = r a n − 1 G e o m e t i c S e q u e n c eĪnd because a n a n − 1 = r, the constant factor r is called the common ratio The constant r that is obtained from dividing any two successive terms of a geometric sequence a n a n − 1 = r. 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”. And adjacent terms, or successive terms, are just two terms in the sequence that come one right after the other. ![]() Well, all we have to do is look at two adjacent terms. 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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