Arithmetic Progression And Formula

Progression in a special type of sequence makes it easy to obtain a formula for it to understand. In mathematics, geometric, arithmetic, and harmonic progression types. Among them, arithmetic progression is used mostly in the sequence pattern. And formulas are to make it easy to understand.

Geometric progression is a series of numbers related by the one common ratio, while this ratio can be positive or negative integration or friction. The geometric progression formula is available, and with the help of the formula, the sum of the geometric progression can find out. Two types of a geometric progression are present will study in this article.

There are three basic types of progression available in mathematics. Progression is a sequence of numbers; calculation of the difference between the sequence numbers with the help of a formula.

Types of progression are as follows:-

1. Arithmetic progression:

It consists of a sequence of numbers, where the variation between the two consecutive numbers has a similar constant number. In the arithmetic progression, there is a possibility to find the nth term with the help of a formula. Arithmetic progression formula:- an= a+(n-1)*d

2. Geometric progression:

 There is a particular sequence in the geometric progression in the geometric progression. One number is different in this sequence, but both have a common ratio. It is termed a geometric sequence or geometric progression. The next number of the sequence is found by multiplying or diving a constant preceding term. It is denoted as a,ar,ar2,ar3,ar4 and arn so on.

Here a represents the number, and r represents the ratio. It needs to keep in mind that.  Division of any succeeding term into preceding terms gives a common ratio.

Geometric progression formula:-

There are several formulas available to solve geometric progression equations.

• It is denoted as a,ar,ar2,ar3,ar4 and arn so on.

• Here a represents the number, and r represents the ratio.

• A geometric progression nth term is Tn = arn-1

• The formula for the common ratio is below:-

Common ratio = r = T_n/ T_n-1

• The formula to calculate the sum of the first n terms of a geometric progression

Sn = a[(1 – rn)/(1 – r)] if r ≠ 1 and r < 1

Sn = a[(rn-1)/(r-1)] if r ≠ 1and r > 1
The nth term is from the geometric progression’s end with the last term l and common ratio r = l/ [r(n – 1)].

• The sum of a geometric progression with infinite terms:-

S∞= a/(1 – r) such that 0 < r < 1.

• Geometric Progression consists of three terms; the middle one is the geometric mean of the other two are considered as terms.

• Geometric Progression consists of three terms a, b and c, then b is the geometric mean of a and c.

Then this will appear as b2 = ac or b =√ac

• Suppose a and r are the first term and common ratio of a finite geometric progression with n terms. Then, the kth term at the geometric progression’s end will be = arn-k.

Types of geometric progression:-

Infinite progression:-

An infinite geometric progression is written as a, ar, ar2, ar3, ……arn-1,…….

This type of series is a, ar, ar2, ar3, ……arn-1,……. is called infinite geometric series.

Finite progression:-

The finite geometric progression is written as a, ar, ar2, ar3,……arn-1. The series a, ar, ar2, ar3,……arn-1 is called finite geometric series. The sum of finite Geometric progression  is  Sn = a[(rn-1)/(r-1)] if r ≠ 1

The Solved Example Of The Geometric Progression Is As Follows:-

If the common ratio of a GP is 3 and the first term is 10. Then write the first five terms of geometric progression.

Answer:- 

In the  question given information,

first term, a = 10

Common ratio, r = 3

As we know the general form of geometric progression for first five

terms is given as follows:-

a, ar, ar2, ar3, ar4

a = 10

ar = 10 x 3 = 30

ar2 = 10 x 32 = 10 x 9 = 90

ar3 = 10 x 33 = 270

ar4 = 10 x 34 = 810

The given first five terms of the geometric progression with

ten as the first term and three as the common ratio is as follows:-

10, 30, 90, 270 and 810

Harmonic progression:-

A harmonic progression or harmonic sequence is a progression that occurs by using the reciprocals of an arithmetic progression. Similarly, harmonic progression has a sequence where each term is harmonic mean and neighbouring are terms.