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    You are at:Home»Pure Math»Algebra»Progressions»Geometric Progression Formulas and Properties & Sum of Geometric Series
    geometric progression formulas and properties | arithmetic and geometric progression question and answers
    Progressions

    Geometric Progression Formulas and Properties & Sum of Geometric Series

    sivaalluriBy sivaalluriSeptember 27, 2019Updated:March 2, 2025No Comments7 Mins Read

    In this session, we explained about the Geometric Progression formulas of nth term, the Sum of the first ‘n’ terms of a G.P., and the Properties of Geometric Progression. Also relation between A.P and G.P.

    Table of Contents

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    • Geometric Progression Formula for nth Term | Properties of Geometric Progression
      • Properties of Geometric progression
      • Geometric Mean

    Geometric Progression Formula for nth Term | Properties of Geometric Progression

    Geometric Progression Definition

    In the sequence, each term is obtained by multiplying a fixed number “r” to the preceding term, except the first term is called Geometric Progression.

    i.e Quantities are said to be in Geometric Progression when they increase or decrease by a constant factor. The constant factor is also called the common ratio.

    Examples:

    1. 30, 60, 120, 240 . . . . . . . .    ( Here fixed multiplying number is “2” )

    2. 1/2, 1/4, 1/8, 1/16, . . . . . . .  ( Here fixed multiplying number is “1/2”)

    3. 10, 2, 0.4, 0.08, . . . . .  . . .  ( Here fixed multiplying number is “0.2”)

    Common Ratio of GP

    The ratio of a term in G.P to its preceding term is called ” Common Ratio” of that G.P.

    i.e The fixed multiplying number “r” in G.P is called common ratio.

    Finding the common ratio of G.P

    The general form of the sequence of a G.P is  a, ar, ar2, ar3, . . . . . . .

    Here, the first term of the sequence is “a” and the common ratio is “r“

    In the above G.P the ratio between any term ( except 1st term) and its preceding term is ” r”

    i.e Common ratio = r  =    \frac{ar}{a} = \frac{ar^2}{ar} = \frac{ar^3}{ar^2} \ . \ . \ . \ . \ . \ .

    Let first term is above GP is a1, second term by a2, . . . . . . .  and nth term by an then

    Common ratio = r  =  \frac{ar_{2}}{ar_{1}} = \frac{ar_{3}}{ar_{2}} = \frac{ar_{4}}{ar_{3}} \ . \ . \ . \ . \ . \ .

    So Common ratio = r =  \frac{ar_{n}}{ar_{n-1}}

    Here, a1, a2, a3 , . . . . . .  an-1, an, an+1 . . . . . . . . . .  is a geometric progression, and each term is non zero , “n” is a natural number and n ≥  2.

    Geometric Progression nth term formula derivation

    Let a1, a2, a3 , . . . . . .  an-1, an, an+1 . . . . . . . . . . is geometric progression and each term is non zero , first term is “a” ,  common ratio is “r” ,  “n” is a natural number and n ≥  2. Then

    1st term is a1 = a

    2nd term is a2 = ar  = ar (2-1)

    3rd term is a3 = ar2 = ar (3-1)

    4th term is a4 = ar3 = ar (4-1)

    5th term is a5 = ar4 = ar (5-1)

    Similarly

    nth term of given GP is  an = arn-1

    Sum of first n terms of a Geometric Progression

    a1, a2, a3 , . . . . . .   is a GP and first term of the sequence is “a” and the common ratio is “r” then the sum of the first n terms of GP is Sn

     S_{n} = \frac{a(1 - r^n)}{1 - r}     if r < 1

     S_{n} = \frac{a(r^n - 1)}{r - 1}     if r > 1

    Sn = na if r = 1

    Sum of infinite G.P is  S_{\alpha} = \frac{a}{1 - r}      If |r | <1

    Properties of Geometric progression

    a) Let  a1, a2, a3 , . . . . . .  an-1, an  are in geometric progression then

    1. If a finite G.P, the product of the terms equidistant from the beginning and end is always the same and is equal to the product of the first and last terms.

    i.e  a2 an-1  = a3 an-2 = a4 an-3  = .  .  .  .  .  . = a1 an

    2.  an ,  an-1, an-2,   . . . . . . . .  a2 ,  a1  are in G.P

    3. xa1, xa2, xa3 , . . . . . . x an-1, xan  are in geometric progression. Here  x ∈  R – {0}

    4. a1n, a2n , a3n, . . . . . .  an-1n , ann are in geometric progression. Here  n ∈  R. i.e if every term of a G.P is raised to the same power, then the resulting series is also a G.P

    5. 1/a1, 1/a2, 1/a3 , . . . . . .  1/an-1, 1/an  are in geometric progression

    b) Let a1, a2, a3 , . . . . . .  an-1, an  & b1, b2, b3 , . . . . . .  bn-1, bn are in geometric progression with common ratio r1 and r2 respectively then,

    1. a1b1, a2b2,  a3b3, , . . . . . .  an-1bn-1, anbn, are in G.P with common ratio r1r2

    2. a1/b1, a2/ b2,  a3/  b3, , . . . . . .  an-1/ bn-1, an/ bn, are in G.P with common ratio r1/ r2

    Geometric Mean

    If three quantities are in geometrical progression, the three middle one is called the geometric mean between the other two.

    a) The geometric mean “G” of any two numbers “a” and “b” then

     G = \sqrt{ab} where a, G , b are in G.P

    b) If a1, a2, a3 , . . . . . .  an-1, an  are “n” numbers, then the Geometric Mean of these numbers is

    G = ( a1 a2 a3  . . . . . .  an-1 an ) 1/n

    c) The ‘n’ numbers G1, G2, G3, . . . . . . .  Gn are said to be G.Ms between ‘a’ and ‘b’. If a, G1, G2, G3, . . . . . . .  Gn, b are in G.P.

    Here common ratio      r = \left( \frac{b}{a} \right)^{\frac{1}{n+1}}

     G_{1} = a \left( \frac{b}{a} \right)^{\frac{1}{n+1}}

     G_{2} = a \left( \frac{b}{a} \right)^{\frac{2}{n+1}}

     G_{3} = a \left( \frac{b}{a} \right)^{\frac{3}{n+1}}

     G_{n} = a \left( \frac{b}{a} \right)^{\frac{n}{n+1}}

    G1 G2 G3. . . . . .  Gn   =  (\sqrt{ab})^n   = ( Geometric Mean of ab)n

    d) If ‘a’ and ‘b’ are two numbers of opposite signs the Geometric mean between them does not exist.

    Relation Between Arithmetic Progression and Geometric Progression

    a) Let a1, a2, a3 , . . . . . .  an-1, an is geometric progression and each term is non zero and non-negative terms then

    log a1, log a2, log a3 , . . . . . .  log an-1, log an are in Arithmetic progression and vice versa.

    b) If a, b, c are consecutive terms of an Arithmetic progression, then xa, xb, xc are the consecutive terms of  geometric progression

    Thanks for reading this article. I hope you liked this article of “ Geometric Progression Formulas and Properties ”. Give feedback and comments, please.

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    Arithmetic progression Geometric Mean Geometric progression geometric progression formula geometric progression tutorial harmonic progression nth term of gp properties of geometric progression sum of gp formula
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