![]() 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, (previous) .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. 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) .(next): arithmetic progression (arithmetic sequence) 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) .Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) . Spiegel: Mathematical Handbook of Formulas and Tables . (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Sum of Arithmetic Progression to $n$ Terms: $3.1.9$ Stegun: Handbook of Mathematical Functions . This is because the word is being used in its adjectival form. ![]() In the context of an arithmetic sequence or arithmetic-geometric sequence, the word arithmetic is pronounced with the stress on the first and third syllables: a-rith- me-tic, rather than on the second syllable: a- rith-me-tic. Let $\sequence \)ĭoubt has recently been cast on the accuracy of the tale about how Carl Friedrich Gauss supposedly discovered this technique at the age of $8$.
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