Geometric Progression Formulas and Properties & Sum of Geometric Series

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

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

General form of sequence of a G.P is  a, ar, ar2, ar3, . . . . . . .

Here first term of sequence is “a” and 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 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 sequence is “a” and common ratio is “r” then sum of 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, a are in geometric progression then

1. If a finite G.P, the product of the terms equidistant from the the beginning and end is always 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 ,  aare in G.P

3. xa1, xa2, xa3 , . . . . . . x an-1, xa 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/a 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, 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, a are “n” numbers then Geometric Mean of these numbers is

G = ( a1 a2 a3  . . . . . .  an-1 a) 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 \ = \ ( \frac{b}{a}) \ \ ^{\frac{1}{n+1}}

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

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

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

G_{n} \ = a \ ( \frac{b}{a}) \ \ ^{\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

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My self Sivaramakrishna Alluri. Thank you for watching my blog friend

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