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 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, 4 red triangles has 1/9 the area of a yellow triangles, or the area of a blue triangle.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.The area of the blue, green, and yellow triangles isĮach of the 48 = 3 4 yellow triangles has 1/9 the area of a green triangle, or the area of a blue triangle.There are three green triangles, so the green and blue triangles have an area ofĮach of the 12 = 3 Each side of the green triangle is exactly 1/3 the length of a side of the blue triangle, and therefore has exactly 1/9 the area of the blue triangle. The fifth iteration of the snowflake is shown below, with its iterations in different colours.Īssume that the one blue triangle as unit area. Now, to derive an expression for the area of our construction at the iteration, let's start with the fifth iteration. The first iteration is blue, the second green, the third yellow, the fourth is red, and the fifth is black (Creative Commons, image from Wikimedia Commons). Absolute Convergence Implies Convergence.The Contrapositive and the Divergence Test.A Motivating Problem for the Alternating Series Test.Example: Integral Test with a Logarithm.A Second Motivating Problem for The Integral Test.A Motivating Problem for The Integral Test.Final Notes on Harmonic and Telescoping Series.
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