Arithmetic Geometric Sequence

Sequence means an order.  Now, the point is what is a sequence and how is it related to the subject of Mathematics. An arithmetic sequence is about numbers that add or subtract.  The geometric progression goes from one term to another and multiplies or divides. The result is a shared value. Ref the figure below, in the arithmetic sequence the difference (d) is always a standard value 7. In the geometric series, the common value is always 2. An arithmetic sequence is about addition (or subtraction) in a set order. A geometric sequence is about multiplying (or division) in a set order.

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Arithmetic Sequence Formula

Let us understand the Arithmetic sequence formula. In the arithmetic sequence, one term goes to the next term by always adding—for example, 1, 2, 3, 4, 5 ….10, and so on. Here, in this sequence, each number moves to the second number by adding (or subtracting 1). Let us take some examples to understand better. 2, 5, 8, 11, 14….is arithmetic sequence as each step adds 3. The same holds for a reverse order (in subtraction). E.g. 7, 3, -1, -5 …is an arithmetic sequence as each step subtracts 4.

It is essential to note that the number that is added or subtracted at each level of an arithmetic sequence is called as the difference (d). The reason is that if you add or subtract (also known as finding the difference), you always get the same common value. Ref fig 2 below

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Geometric Sequence Formula

A geometric sequence or geometric series is a geometric order. We obtain results by multiplying the terms of a geometric sequence. In simple words, a geometric sequence moves from one term to the next by always multiplying (or by division) by the same common value or number. For example, 2, 4, 8, 16, 32 … is a geometric series. The reason being that each step multiplies by two. Let us take another example, 81, 27, 9, 3, 1 …it is a geometric sequence as each step divides (or multiplies) by the number 3.

It is essential to note that the common number that multiplies or divides at each step of a geometric sequence is called the ratio r. It is due to the fact that if you divide or find the ratio of the successive terms, you get a common or standard ratio. Ref Fig.3 below.

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Arithmetic Geometric Progression 

An arithmetic progression is a series or sequence of numbers in which each term is derived from the next term, by adding or subtracting a fixed or common number called the common difference d. For instance, the series, 9, 6, 3, 0,-3 and so on, is an arithmetic progression with -3 as the standard difference. The progression -3, 0, 3, 6, 9 is an example of Arithmetic Progression (AP) that has three as the common difference d.

The established form of AP is a, a + d, a + 2d, a + 3d and ….so on. The nth term will be

an = a + (n -1)d  

A geometric progression is a series in which each term is obtained by a multiplication or division of the next term, by a fixed or common number. For instance, the series 8, 4, -2, 1, -1/2 … is a Geometric Progression (GP) for which -1/2 is the common ratio.  

The established form of GP is a, ar, ar2, ar3, ar4….and so on. The nth term will be   an = ar (n-1)  

Example -1

Work out the common difference and the next term of the following series: 

3, 11, 19, 27, 35 … 

We have to find the common difference d. you can pick up any pair. Let us start with subtractions. 

11 - 3 = 8 

19 -11 = 8 

27 -19 = 8 

35 - 27 = 8 

Throughout, the difference is 8, so the common difference is 8.

We have 5 terms. We have to find the next or the 6th term. We can find out by adding the common difference to the fifth term, 35 + 8 = 43. 

sixth term = 43 and common difference =8

Example -2 

Find the common ratio and the 7th term of the following sequence 

2/9, 2/3, 2, 6, 18. 

We will take 6/2 = 3 and 18/6 = 3, the common ratio is 3, so r = 3  

We have 5 terms, we have to find the 6th, then the 7th term. 

So, a6 = 18 x 3 = 54 

 a7 = 54 x 3 = 162, the answer is common ratio r = 3 and seventh term = 162. 

FAQ (Frequently Asked Questions)

1. Can arithmetic and geometric sequence be the same? 

There are specific differences as well as similarities between arithmetic and geometric sequences. Arithmetic sequences or series follow terms by adding. Geometric series or sequences follow terms by multiplications. The similarities between AP and GP is that they follow a definite pattern. This pattern cannot be broken.  For example, (2, 4, 8), (9, 18, 36), (8, 16, 32) etc. An arithmetic sequence is obtained by adding or subtracting the common difference d. Geometric sequences are obtained by multiplying (or dividing) the previous term by a common ratio r.  When we find out the common difference and the common ratio, it is easy to find the order.

2. What is the use of arithmetic progression?

In the subject of mathematics, arithmetic progression is a series of numbers. The numbers follow a fixed pattern that has a common difference between them. The difference between the numbers is always constant.  The difference here means that you minus the second number from the first number. For instance, in the sequence 2, 4, 6, 8, 10 … the common difference is 2. The clear advantage of AP is that it can be of use to determine future trends and the statistics. We can quickly obtain the next figures by calculating the nth term in the AP sequence.