![]() ![]() 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. Value of an Annuity with Interest Compounded n n Times a Yearĭetermine if each sequence is geometric.Apply Geometric Sequences and Series in the Real World.Find the Sum of an Infinite Geometric Series.Sum of the First n Terms of a Geometric Series.Find the Sum of the First n Terms of a Geometric Sequence. ![]()
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