Arithmetic and Geometric Progressions Mathematics

Introduction

The number sequences of arithmetic and geometric progressions are widely used in several engineering applications. That includes sorting algorithms, computer data structure, audio compression, financial engineering, and architectural engineering. People use arithmetic progressions in the reproductive cycle of bacteria and simulative engineering. These are also used in daily life like completing the pattern of objects, speed of an aircraft, calculating sales, production, simple interest, etc. Geometric progressions are mostly used to calculate compound interest. Before understanding arithmetic and geometric progressions, we need to understand sequences and series.

Sequence

A sequence refers to a set of numbers that are written in a particular order. For example,

2, 4, 6, 8, 10, …

Here, we can see that we have a sequence of even numbers. To put this in another way, we start with the number 2, which is an even number, and then obtain each successive number by adding 2 to the next number.

Here is another example of a sequence.

1, 3, 5, 7, 9, …

This is a sequence of odd numbers.

Another sequence is as follows,

1, -1, 1, -1, 1, …

This is a sequence of numbers that alternate between 1 and -1. In each of these cases, the dots at the end indicate an infinite sequence that goes on forever.

Sometimes, we also get finite sequences. One such example is as follows:

1, 3, 5, 9.

Sometimes, we use different terms for a sequence. For instance, we use u1 for the first number, u2 for the second number, and so on. Moreover, we consider the nth term as un.

Sum

The sum is always obtained from a sequence. That is done by adding all the terms together. Firstly, we will take the example of the following sequence:

x1, x2, x3, x4, …xn.

So, the sum that we obtain from this sequence is

x1 + x2 + x3 + x4 + … + xn.

The sum of these n terms is written as Sn. 

Arithmetic progressions

Arithmetic progressions refer to a sequence of numbers that differ from the preceding number by a constant quantity. In other words, arithmetic progressions or APs are simply the relations between numbers. For this, let us consider the two common sequences below:

Sequence 1: 1, 3, 5, 7, …

Sequence 2: 0, 10, 20, 30, 40, …

It is very easy to understand how the above sequences are formed. Each of them begins with the first particular term, and then a fixed value is added to the same to get the successive term. In sequence 1, we add the fixed number 2 while in sequence 2 we add the fixed number 10. So, the difference between the consecutive terms in each of the sequences is a constant. The same can be done in reverse by subtracting a constant in a series. For example,

8, 5, 2, -1, -4, …

So, the difference between the above consecutive terms is -3. Such sequences of properties are termed arithmetic progressions (APs).

Sum of an Arithmetic Series

Sometimes, there is a need to add all the terms of a sequence. So, if we want to add the first n terms of an arithmetic progression, we will get the following:

Sn = x + (x + z) + (x+ 2z) + . . . (x +(n-1)z).

In this series, we have added the n terms of the sequence. This is what mathematics calls an arithmetic series. We can find the sum easily by using this trick. We will now write down the same series in reverse order.

Sn = x+(n-1)z + x+(n-2)z…….+ x 

So, the sum of the terms of an arithmetic progression gives us an arithmetic series. Here is the formula:

Sn = 1/2 n (2a + (n − 1) d).

Here, a is the starting value,

d is a common difference.

Geometric Progressions

Geometric progressions refer to the progression of numbers having a constant ratio between each of them and the one before.

Let us consider the sequence below to understand better:

2, 6, 18, 54, …

Here, each term of the sequence is thrice the previous term. Let us consider another sequence:

1, -2, 4, -8, …

Here, each term is -2 times the previous one. These sequences are termed geometric progressions or GPs.

Sum of a Geometric Series

If we want to find the sum of the first n terms of a geometric progression, here is what we get

Sn = x + xy + xy2 + xy3 + . . . + xyn-1.

x is the first term and y is the common ratio.

So, the sum of terms of a geometric progression is what gives us a geometric series.

Conclusion

The above study material notes on arithmetic and geometric progressions have taught us the formulae, sum, and other interesting facts related to these progressions. You can solve arithmetic and geometric progressions easily after having a look at the series or sequence patterns. Both these progressions explore particular types of sequence that make them different and unique from each other. Talking about arithmetic sequence, it is a set of numbers in which each of the new phrases differs from the previous term by a fixed amount. A geometric sequence, on the other hand, is a series of integers in which every element is obtained by multiplying the preceding number with a constant factor.