We can see from the given explicit formula that \(r=2\). Find \(a_1\) by substituting \(k=1\) into the given explicit formula.Therefore, a convergent geometric series An infinite geometric series where | r | < 1 whose sum is given by the formula: S ∞ = a 1 1 − r. If | r | < 1 then the limit of the partial sums as n approaches infinity exists and we can write, S n = a 1 ( 1 − r n ) 1 − r = a 1 1 − r ( 1 − r n ) Consider the nth partial sum of any geometric sequence, Try it in your head with a simple series, such as whole numbers from 1 to 10. This is read, “the limit of ( 1 − r n ) as n approaches infinity equals 1.” While this gives a preview of what is to come in your continuing study of mathematics, at this point we are concerned with developing a formula for special infinite geometric series. The fundamental insight that originally led to the creation of this formula probably started with the observation that the sum of the first term and last term in an arithmetic series is always the same as the sum of the 2nd and 2nd-to-last, 3rd and 3rd-to-last, etc. Lim n → ∞ ( 1 − r n ) = 1 w h e r e | r | < 1 This illustrates the idea of a limit, an important concept used extensively in higher-level mathematics, which is expressed using the following notation: Here we can see that this factor gets closer and closer to 1 for increasingly larger values of n. If the common ratio r of an infinite geometric sequence is a fraction where | r | < 1 (that is − 1 < r < 1), then the factor ( 1 − r n ) found in the formula for the nth partial sum tends toward 1 as n increases. 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 The sum to infinity of a geometric series is given by the formula. Multiplying both sides by r we can write, It is only possible to calculate the sum to infinity for geometric series that. S n = a 1 + a 1 r + a 1 r 2 + … + a 1 r n − 1 But it is easier to use this Rule: x n n (n+1)/2. Therefore, we next develop a formula that can be used to calculate the sum of the first n terms of any geometric sequence. The Triangular Number Sequence is generated from a pattern of dots which form a triangle: By adding another row of dots and counting all the dots we can find the next number of the 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, 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.
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