Also, read |

A.P Formula |

H.P Formula |

# Geometric Progression:

A series of numbers where each consecutive term is obtained by multiplying the previous term by a constant number every time then the series is said to be in **Geometric Progression(G.P**). We can also say that when two consecutive term in a series is divided and the quotient remains same every time then the series is said to be in Geometric Progression. Geometric progression (G.P) is also known as **Geometric Sequence**.

The constant (quotient) is said to be **Common Ratio.**

**Terms used in Geometric Progression**

**a – **first term or scale factor

**r – **Common Ratio

**n –** number of terms

**a _{n} –** n

^{th}term

**S _{n }– **Sum of n terms of G.P

**S _{∞} – **Sum of infinite

**(**

**∞)**terms of G.

**Geometric Progression is expressed as**

**a, ar, ar ^{2}, ar^{3}, ….. Where r**

**≠ 0**

**Common Ratio Rules:**

If common ratio is

,*Negative***the series will be positive and negative****consecutively.**

**For example:
**1, -2, 4, -8, 16, -32… – in this series common ratio is -2 and the first term is 1.

,*Greater than 1***the series will move exponentially towards positive (+ve) infinity**.

**For Example:
**1, 5, 25, 125, 625 … – in this series common ratio is 5.

*Less than -1***, the series will move exponentially towards negative (-ve) infinity.**

**For Example:
**1, -5, 25, -125, 625, -3125, 15625, … – in this series common ratio is -5.

,*Between 1 and -1***the series will be exponential move towards zero**.

**For Example:**

4, 2, 1, 0.5, 0.25, 0.125, 0.0625 … – in this series common ratio is \(\frac{1}{2}\)

4, -2, 1, -0.5, 0.25, -0.125, 0.0625 … – in this series common ratio is – \(\frac{1}{2}\)

,*Zero***the series will continue to remain at zero**.

**For Example:**4, 0, 0, 0, 0 … – in this series the common ratio is 0 and the first term is 4.

**Formula related to G.P**

** a _{n} = ar^{n-1}**

**S _{n} = \(\frac{a(1-r^{n})}{1-r}\) **

**r**

**≠ 1**

**S _{n} =**

**a.n**if

**r**

**= 1**

**S _{∞} = \(\frac{a}{1-r}\) **provided |r|< 1

**Properties of G.P**

**a ^{2}_{k} = a_{k-1}**

**⋅**

**a**

a

_{k+1}a

_{1}**⋅**

**a**

_{n}= a_{2}**⋅**

**a**

_{n-1}=…= a_{k}**⋅**

**a**

_{n-k+1}**Examples related to G.P:**

Q. ** If 2, 4, 8… are in G.P. Find the 6-**th

*term?***Solution:
**

**a**

_{n}**= a**

_{1}

**⋅**

**r**

^{n-1}**a**

_{6}= 2 ⋅ 2

^{6-1}= 2 ⋅ 32 = 64

Q.* Find the ‘a’ and the ‘r’ of a G.P if*

*a _{5}– a_{1}= 15*

*a*_{4}– a_{2}= 6**Solution:
**There are two G.P.

The first one has a =1 and common ratio r= 2

In second term is -16, 1/2

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