In this session explained about **Geometric Progression formulas** of n^{th} 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 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

General form of sequence of a G.P is a, ar, ar_{2}, ar_{3}, . . . . . . .

Here first term of sequence is “**a**” and 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 =

Let first term is above GP is a_{1}, second term by a_{2}, . . . . . . . and n^{th }term by a_{n} then

Common ratio = r =

So Common ratio = r =

Here a_{1}, a_{2}, a_{3} , . . . . . . a_{n-1}, a_{n}, a_{n+1} . . . . . . . . . . is *geometric progression* and each term is non zero , “n” is a natural number and n ≥ 2.

**Geometric Progression n ^{th} term formula derivation**

Let a_{1}, a_{2}, a_{3} , . . . . . . 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 a_{1} = a

2^{nd} term is a_{2} = ar = ar ^{(2-1)}

3^{rd} term is a_{3} = ar^{2} = ar ^{(3-1)}

4^{th} term is a_{4} = ar^{3} = ar ^{(4-1)}

5^{th} term is a_{5} = ar^{4} = 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**

a_{1}, a_{2}, a_{3} , . . . . . . 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

if r < 1

if r > 1

Sn = na if r = 1

Sum of infinite G.P is If |r | <1

### Properties of Geometric progression

**a**) Let ^{ }a_{1}, a_{2}, a_{3} , . . . . . . a_{n-1}, a_{n } 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 a_{2 }a_{n-1 }= a_{3 }a_{n-2 }= a_{4 }a_{n-3 }= . . . . . . = a_{1 }a_{n}

2. a_{n} , a_{n-1, }a_{n-2, }. . . . . . . . a_{2 , }a_{1 }are in G.P

3. xa_{1}, xa_{2}, xa_{3} , . . . . . . x a_{n-1}, xa_{n } are in geometric progression. Here x ∈ R – {0}

4. a_{1}^{n}, a_{2}^{n} , a_{3}^{n}, . . . . . . a_{n-1}^{n} , a_{n}^{n }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/a_{1}, 1/a_{2}, 1/a_{3} , . . . . . . 1/a_{n-1}, 1/a_{n } are in geometric progression

**b)** Let a_{1}, a_{2}, a_{3} , . . . . . . a_{n-1}, a_{n} & b_{1}, b_{2}, b_{3} , . . . . . . b_{n-1}, b_{n} are in geometric progression with common ratio r_{1} and r_{2} respectively then,

1. a_{1}b_{1}, a_{2}b_{2}, a_{3}b_{3}, , . . . . . . a_{n-1}b_{n-1}, a_{n}b_{n}, are in G.P with common ratio r_{1}r_{2}

2. a_{1/}b_{1}, a_{2/ }b_{2}, a_{3/ }b_{3}, , . . . . . . a_{n-1/ }b_{n-1}, a_{n/ }b_{n}, are in G.P with common ratio r_{1/ }r_{2}

### 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

Where a, G , b are in G.P

**b)** If a_{1}, a_{2}, a_{3} , . . . . . . a_{n-1}, a_{n } are “n” numbers then Geometric Mean of these numbers is

G = ( a_{1} a_{2} a_{3} . . . . . . a_{n-1} a_{n }) ^{1/n}

**c)** The ‘n’ numbers G_{1}, G_{2}, G_{3}, . . . . . . . G_{n} are said to be G.Ms between ‘a’ and ‘b’. If a, G_{1}, G_{2}, G_{3}, . . . . . . . G_{n}, b are in G.P.

Here common ratio

G_{1} G_{2} G_{3}. . . . . . G_{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 a_{1}, a_{2}, a_{3} , . . . . . . a_{n-1}, a_{n} is geometric progression and each term is non zero and non negative terms then

log a_{1}, log a_{2}, log a_{3} , . . . . . . log a_{n-1}, log a_{n }are 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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