![]() 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, On the other hand, the practical application of geometric sequence is to find out population growth, interest, etc.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. Further, an arithmetic sequence can be used find out savings, cost, final increment, etc. Hence, with the above discussion, it would be clear that there is a huge difference between the two types of sequences. The infinite arithmetic sequences, diverge while the infinite geometric sequences converge or diverge, as the case may be. ![]() As against this, the variation in the elements of the sequence is exponential. In an arithmetic sequence, the variation in the members of the sequence is linear.As opposed to, geometric sequence, wherein the new term is found by multiplying or dividing a fixed value from the previous term. In an arithmetic sequence, the new term is obtained by adding or subtracting a fixed value to/from the preceding term.On the contrary, when there is a common ratio between successive terms, represented by ‘r’, the sequence is said to be geometric. A sequence can be arithmetic, when there is a common difference between successive terms, indicated as ‘d’.A set of numbers wherein each element after the first is obtained by multiplying the preceding number by a constant factor, is known as Geometric Sequence. As a list of numbers, in which each new term differs from a preceding term by a constant quantity, is Arithmetic Sequence.The following points are noteworthy so far as the difference between arithmetic and geometric sequence is concerned: Key Differences Between Arithmetic and Geometric Sequence Geometric Sequence is a set of numbers wherein each element after the first is obtained by multiplying the preceding number by a constant factor.Ĭommon Difference between successive terms. Content: Arithmetic Sequence Vs Geometric SequenceĪrithmetic Sequence is described as a list of numbers, in which each new term differs from a preceding term by a constant quantity. Here, in this article we are going to discuss the significant differences between arithmetic and geometric sequence. ![]() In an arithmetic sequence, the terms can be obtained by adding or subtracting a constant to the preceding term, wherein in case of geometric progression each term is obtained by multiplying or dividing a constant to the preceding term. On the other hand, if the consecutive terms are in a constant ratio, the sequence is geometric.
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