Geometric Series Formula
The geometric series formula is used to calculate the sum of an infinite geometric sequence or the sum of a finite geometric sequence
The geometric series formula is used to calculate the sum of an infinite geometric sequence or the sum of a finite geometric sequence.
For an infinite geometric series, where the first term is “a” and the common ratio is “r” (with |r| < 1), the formula is: S = a / (1 - r) Here, S represents the sum of the infinite geometric series. For example, consider the series 3 + 1 + 1/3 + 1/9 + ... where a = 3 and r = 1/3. Using the formula, we can find the sum: S = 3 / (1 - 1/3) = 3 / (2/3) = 9/2 So, the sum of the infinite geometric series is 9/2. For a finite geometric series, which consists of "n" terms, the formula is slightly different and is given by: Sn = a * ((1 - r^n) / (1 - r)) Here, Sn represents the sum of the finite geometric series. For example, let's consider the series 2 + 4 + 8 + 16 + 32. This series has 5 terms, so n = 5. In this case, a = 2 and r = 4/2 = 2. Using the formula, we can find the sum: S5 = 2 * ((1 - 2^5) / (1 - 2)) = 2 * ((1 - 32) / (-1)) = 2 * (-31) = -62 So, the sum of the finite geometric series is -62. It is important to note that the formula for the sum of a finite geometric series is only valid when |r| < 1. If the common ratio is greater than or equal to 1, the series diverges and does not have a finite sum.
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