Geometric Progression Formulas and Properties & Sum of Geometric Series

In this session, we explained about the Geometric Progression formulas of n^th 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

Geometric Progression Formula for n^th 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 1^st 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 n^thterm by a_n 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 , . . . . . . a_n-1, a_n, a_n+1 . . . . . . . . . . is a geometric progression, and each term is non zero , “n” is a natural number and n ≥ 2.

Geometric Progression n^th term formula derivation

Let a₁, a₂, a₃ , . . . . . . a_n-1, a_n, a_n+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

1^st term is a₁ = a

2^nd term is a₂ = ar = ar ^(2-1)

3^rd term is a₃ = ar² = ar ^(3-1)

4^th term is a₄ = ar³ = ar ^(4-1)

5^th term is a₅ = ar⁴ = ar ^(5-1)

Similarly

n^th term of given GP is a_n = ar^n-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 , . . . . . . a_n-1, a_n 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 a₂a_n-1= a₃a_n-2= a₄a_n-3= . . . . . . = a₁a_n

2. a_n , a_n-1,a_n-2,. . . . . . . . a_{2 ,}a₁are in G.P

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

4. a₁ⁿ, a₂ⁿ , a₃ⁿ, . . . . . . a_n-1ⁿ , a_nⁿare 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/a_n-1, 1/a_n are in geometric progression

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

1. a₁b₁, a₂b₂, a₃b₃, , . . . . . . a_n-1b_n-1, a_nb_n, are in G.P with common ratio r₁r₂

2. a_1/b₁, a_2/b₂, a_3/b₃, , . . . . . . a_n-1/b_n-1, a_n/b_n, are in G.P with common ratio r_1/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 , . . . . . . a_n-1, a_n are “n” numbers, then the Geometric Mean of these numbers is

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

c) The ‘n’ numbers G₁, G₂, G₃, . . . . . . . G_n are said to be G.Ms between ‘a’ and ‘b’. If a, G₁, G₂, G₃, . . . . . . . G_n, 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. . . . . . G_n= $(\sqrt{ab})^n$ = ( 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 , . . . . . . a_n-1, a_n is geometric progression and each term is non zero and non-negative terms then

log a₁, log a₂, log a₃ , . . . . . . log a_n-1, log a_nare in Arithmetic progression and vice versa.

b) If a, b, c are consecutive terms of an Arithmetic progression, then x^a, x^b, x^c are the consecutive terms of geometric progression

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