![]() Let’s look at an infinite geometric series whose common ratio is a fraction less than one,ġ 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + … 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + …. We cannot find a sum of an infinite geometric series when | r | ≥ 1. This is true when | r | ≥ 1 | r | ≥ 1 and we call the series divergent. Let’s look at a few partial sums for this series. Each term gets larger and larger so it makes sense that the sum of the infinite number of terms gets larger. Let’s look at the infinite geometric series 3 + 6 + 12 + 24 + 48 + 96 + …. But how do we find the sum of an infinite sum? We know how to find the sum of the first n terms of a geometric series using the formula, S n = a 1 ( 1 − r n ) 1 − r. S n = a 1 ( 1 − r n ) 1 − r S n = a 1 ( 1 − r n ) 1 − rĪn infinite geometric series is an infinite sum whose first term is a 1 a 1 and common ratio is r and is writtenĪ 1 + a 1 r + a 1 r 2 + … + a 1 r n − 1 + … a 1 + a 1 r + a 1 r 2 + … + a 1 r n − 1 + … S n ( 1 − r ) = a 1 ( 1 − r n ) S n ( 1 − r ) = a 1 ( 1 − r n )ĭivide both sides by ( 1 − r ). S n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 r S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 + a 1 r n _ S n − r S n = a 1 −a 1 r n S n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 r S n = a 1 r + a 1 r 2 + a 1 r 3 + … + a 1 r n − 1 + a 1 r n _ S n − r S n = a 1 −a 1 r n We will see that when we subtract, all but the first term of the top equation and the last term of the bottom equation subtract to zero. + a 1 r n r S n = a 1 r + a 1 r 2 + a 1 r 3 +. Let’s also multiply both sides of the equation by r. + a 1 r n − 1 S n = a 1 + a 1 r + a 1 r 2 +. We can write this sum by starting with the first term, a 1, a 1, and keep multiplying by r to get the next term as: The sum, S n, S n, of the first n terms of a geometric sequence is written as S n = a 1 + a 2 + a 3 +. We will now do the same for geometric sequences. We found the sum of both general sequences and arithmetic sequence. įind the Sum of the First n Terms of a Geometric Sequence Ⓑ Find the ratio of the consecutive terms. To determine if the sequence is geometric, we find the ratio of the consecutive terms shown. \)ĭetermine if each sequence is geometric. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. We recommend using aĪuthors: Lynn Marecek, Andrea Honeycutt Mathis Use the information below to generate a citation. Then you must include on every digital page view the following attribution: ![]() If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the ![]() In each term, the number of times a 1 a 1 is multiplied by r is one less than the number of the term. r or r 3 r 3) and in the fifth term, the a 1 a 1 is multiplied by r four times.In the fourth term, the a 1 a 1 is multiplied by r three times ( r In the third term, the a 1 a 1 is multiplied by r two times ( r In the second term, the a 1 a 1 is multiplied by r. ![]() The first term, a 1, a 1, is not multiplied by any r. We will then look for a pattern.Īs we look for a pattern in the five terms above, we see that each of the terms starts with a 1. Let’s write the first few terms of the sequence where the first term is a 1 a 1 and the common ratio is r. Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence. Find the General Term ( nth Term) of a Geometric Sequence Write the first five terms of the sequence where the first term is 6 and the common ratio is r = −4.
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