What is a Geometric Progression?

What is a Geometric Progression?

Geometric Progression (G.P.)

Definition:
A progression is called a G.P. if the ratio of its each term to its previous term is always constant. This constant ratio is called its common ratio and it is generally denoted by r.

Example:

General term of a G.P.

(1) We know that a, ar, ar², ar³, ……. arⁿ⁻¹ is a sequence of G.P.
Here, the first term is ‘a’ and the common ratio is ‘r’.
The general term or n^th term of a G.P. is T_n = arⁿ⁻¹.

(2) p^th term from the end of a finite G.P. : If G.P. consists of ‘n’ terms, p^th term from the end = (n – p + 1)^th term from the beginning = ar^n–p.
Also, the p^th term from the end of a G.P. with last term l and common ratio r is

Selection of terms in a G.P.

(1) When the product is given, the following way is adopted in selecting certain number of terms :

(2) When the product is not given, then the following way is adopted in selection of terms

Sum of first ‘n’ terms of a G.P.

If a be the first term, r the common ratio, then sum of first n terms of a G.P. is given by

Sum of infinite terms of a G.P.

Geometric mean

Properties of G.P.

1. If all the terms of a G.P. be multiplied or divided by the same non-zero constant, then it remains a G.P., with the same common ratio.
2. The reciprocal of the terms of a given G.P. form a G.P. with common ratio as reciprocal of the common ratio of the original G.P.
3. If each term of a G.P. with common ratio r be raised to the same power k, the resulting sequence also forms a G.P. with common ratio r^k.
4. In a finite G.P., the product of terms equidistant from the beginning and the end is always the same and is equal to the product of the first and last term. i.e., if a₁, a₂, a₃, …… a_n be in G.P.
   Then a₁a_n = a₂a_n-1 = a₃a_n-2 = a₄a_n-3 = a_r.a_n-r+1
5. If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed also forms a G.P.
6. If a₁, a₂, a₃, …… a_n is a G.P. of non-zero, non-negative terms, then log a₁, log a₂, log a₃, …… log a_n is an A.P. and vice-versa.
7. Three non-zero numbers a, b, c are in G.P., iff b² = ac.
8. If first term of a G.P. of n terms is a and last term is l, then the product of all terms of the G.P. is (al)^n/2.
9. If there be n quantities in G.P. whose common ratio is r and S_m denotes the sum of the first m terms, then the sum of their product taken two by two is \(\frac { r }{ r+1 } { S }_{ n }{ S }_{ n-1 }\).
10. If a^x1, a^x2, a^x3, ……, a^xn are in G.P., then x₁, x₂, x₃ …… x_n will be are in A.P.

Geometric Progression Problems with Solutions

1. If the and  terms of  a G.P. be  respectively, then the relation between  is

Solution:

2.

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3.

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4.

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5.

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6.

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7.

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8.

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