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A geometric progression, also defined as a geometric sequence, is a non-zero number sequence where each term after the first is identified by multiplying it by a fixed, non-zero number called the common ratio.
Geometric progression
Geometric progression, or G.P., belongs to the category of progressions, that are mathematical sequences in which each succeeding term is formed by multiplying the preceding term with a specific fixed number. The term ‘sequence’ refers to a collection of objects that are organised in such a way that each of them can be identified individually. A progression is when a sequence in a mathematical application follows a specific pattern which can be derived through calculations.
Geometric Progression (G.P.): Sequence and Series
Suppose the components A1, A2, A3,……, An to form a finite sequence with ‘n’ terms. The sequence A1+A2+A3+……+An is identified as the series associated with this sequence in this scenario.
Geometric progression Definition and General Form
A geometric progression, or G.P., occurs when the ratio of a particular term (except the first) to its preceding term is the same and remains constant throughout the length of the sequence. In simple contexts, every term in this equation has a constant ratio to the term before it. So, to get the next term in a given G.P., To get the previous term, multiply by the constant ratio, while to get the previous term, divide by the same constant ratio.
Components of Geometric progression
The following are the major characteristics of a G.P. :
The ‘first term,’ that is a G.P.’s 1st term;
The specialised constant factor which marks the ratio between the successive terms of a G.P., computed by dividing the 2 successive terms; the ‘common ratio,’ that can be a positive or negative integer;
The ‘number of terms’, usually defined by ‘n’; the G.P.’s ‘general term,’ that is the nth term.
The nth term can be identified by using the formula to find the 1st term of a specified G.P.
An=Ar n-1
Here,
The 1st term is A, the standard ratio is r, and the total number of terms is n.
Types of Geometric Progression.
The 2 basic forms of G.P. are finite and infinite depending on the number of terms present in a particular progression sequence.
A finite G.P. has a finite number of terms and a specified last term by definition. The general expression A, Ar, Ar 2 , Ar 3,…… can be used to describe it Ar n-1.
There are an infinite number of terms in an infinite G.P., and the last term is not defined clearly. It can be defined as A, Ar, Ar 2 , Ar 3,…… Ar n-1…….
Sample questions on Geometric Progression for CAT Exams
Solved examples of Geometric Progression
Examples 1: 2,4,8,16,……….. is a geometric progression, with 1024 becoming the nth term. Calculate the value of n.
Sol: The common ratio is clearly 2 and the first term, a = 2
The nth term, xn=xr n-1, is known.
As a consequence, 1024 =2 ×2 n-1
2 n-1= 1024/2
2 n-1=512
2 n-1=29
n-1 = 9
The required value is n = 10.
Examples 2: A girl has 2 parents, 4 grandparents, and 8 great grandparents, among other relatives. Compute the total number of her forefathers and mothers over the last five generations.
Sol: We recognise that our forefathers are putting together a G.P. in which,
The first term is 2, the common ratio is 2, and the number of generations is 5.
Now, applying the sum formula for a G.P.’s n terms, i.e.
x(rn-1) = Sn (r-1)
S5= 2(25– 1) = 62 is the result. As a consequence, the total number of her ancestors is 62.
Examples 3: Find the 12th term of a G.P. if 2,4,8,16,…. is a G.P.
Sol: The nth term of GP is equal to:
A12=2 x 2 12-1
2 x 211= 4096
Important Notes on Geometric Progression
The common ratio is multiplied by the preceding term to determine each subsequent term in a geometric progression.
An=Ar n-1 is the formula for nth term of a geometric progression with a first term and a common ratio of r.
In GP, where the first term is a and the common ratio is r, the formula for calculating the sum of n terms is: Sn = [a(1-rn)] / (1-r).
Sn = a/(1-r), where |r<|1 is the sum of infinite GP formulas.
Conclusion
A geometric progression (GP) has a constant ratio between each term and the one before it. It’s a unique kind of progression. We must multiply with a fixed term referred as the common ratio every time we would like to find the next term in the geometric progression, and we must divide the term with the same common ratio every time we want to find the preceding term in the progression.
