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, This work is licensed under a Creative Commons Attribution 4.0 License.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. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio. We can divide any term in the sequence by the previous term. The common ratio is also the base of an exponential function as shown in Figure 2ĭo we have to divide the second term by the first term to find the common ratio? The sequence of data points follows an exponential pattern. Substitute the common ratio into the recursive formula for geometric sequences and define. The common ratio can be found by dividing the second term by the first term. Write a recursive formula for the following geometric sequence. Substitute the common ratio into the recursive formula for a geometric sequence.ģ Using Recursive Formulas for Geometric Sequences.Find the common ratio by dividing any term by the preceding term.Given the first several terms of a geometric sequence, write its recursive formula. The recursive formula for a geometric sequence with common ratio and first term is Recursive Formula for a Geometric Sequence For example, suppose the common ratio is 9. Each term is the product of the common ratio and the Allows us to find any term of a geometric sequence by using the
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