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Learn about arithmetic progression as well. The common ratio multiplied by each term to get the next term is not zero. A Geometric sequence would be 2, 4, 8, 16, 32, 64…, where the common ratio is 2.
A geometric progression or sequence is a succession in which each term differs from another by a common ratio. When we multiply a constant (that is not zero) by the preceding term, we get the following term in the sequence. It is represented by the characters a, ar, ar2, ar3, ar4, and so on.
Where a represents the first phrase and r represents the common ratio.
It should be noticed that dividing every next phrase by its preceding term yields a result equal to the common ratio.
If we divide the third term by the second term, we get:
r = ar2/ar
Similarly, ar3/ar2 Equals r.
r = ar4/ar3
General Form of Geometric Progression
Geometric Progression takes the following general form: a, ar, ar2, ar3, ar4,…, arn-1.
Where a = the first term
r denotes the common ratio.
nth term = arn-1
nth Term of Geometric Progression
For a Geometric Sequence, let a be the first term and r be the common ratio.
Then comes the second word, a2 = a*r = ar.
The third term, a3 = a2*r = ar*r = ar2
Similarly, an = arn-1 for the nth term.
As a result, the formula for determining the nth term of GP is: a = tn = arn-1.
The nth term is the final term of finite GP.
Sum of N term of GP
Suppose a, ar, ar2, ar3,……arn-1 is the given Geometric Progression.
The sum of n terms of GP is then given by:
Sn = a + ar + ar2 + ar3 +…+ arn-1
The formula to find the sum of n terms in a GP is:
Sn = a[(rn – 1)/(r – 1)] if r ≠ 1 and r > 1
Where
a is the first term
r is the common ratio
n is the number of terms
Also, if the common ratio is equal to 1, then the sum of the geometric progression is given by:
Sn = na if r = 1
Properties of Geometric Progression (GP)
The key features of a GP are:
Three non-zero terms a, b, c are in GP if and only if b2 = ac
In a GP,
Three consecutive terms can be considered as a/r, a, ar
Four consecutive terms can be taken as a/r3, a/r, ar, ar3
Five consecutive terms can be taken as a/r2, a/r, a, ar, ar2
- The product of the terms equidistant from the beginning and end of a finite GP is the same. That means, t1.tn = t2.tn-1 = t3.tn-2 = …..
If a GP’s terms are multiplied or divided by a non-zero constant, the resulting sequence is also a GP with the same common ratio.
The product and quotient of two GP’s is again a GP
If each term of a GP is raised to the power by the same non-zero quantity, the resultant sequence is also a GP
If a1, a2, a3,… is a GP of positive terms then log a1, log a2, log a3,… is an AP (arithmetic progression) and vice versa
Types of Geometric Progression
Based on the number of terms, geometric progression can be classified into two types. They are as follows:
Finite geometric progression (Finite GP)
Infinite geometric progression (Infinite GP)
These two GPs are explained below with their representations and the formulas to find the sum.
Finite Geometric Progression
The terms of a finite G.P. can be written as a, ar, ar2, ar3,……arn-1
a, ar, ar2, ar3,……arn-1is called finite geometric series.
The sum of a finite Geometric progression series is given by:
Sn = a[(rn – 1)/(r – 1)] if r ≠ 1 and r > 1
Infinite Geometric Progression
Terms of an infinite G.P. can be written as a, ar, ar2, ar3, ……arn-1,…….
a, ar, ar2, ar3, ……arn-1,……. is called infinite geometric series.
The sum of an infinite geometric progression series is given by:
This is called the geometric progression formula for the sum to infinity.
Conclusion
A geometric progression is a sort of progression in which the succeeding terms have the same constant ratio, known as the common ratio. It is also referred to as GP. The GP is commonly represented as a, ar, ar2…., where an is the first term and r is the progression’s common ratio. The common ratio can have both positive and negative values. To find the terms of a geometric series, we only need the beginning term and the constant ratio.
