In this section, we are going to learn about geometric progression.

**What is geometric progression ?**

A Geometric Progression is a sequence in which each term is obtained by multiplying a fixed non-zero number to the preceding term except the first term.

The fixed number is called common ratio. The common ratio is usually denoted by r.

General form of geometric progression :

The numbers of the form

a, ar, ar^{2}, ...ar^{n-1}

is called a Geometric Progression.

The number ‘a’ is called the first term and number ‘r’ is called the common ratio.

To find the common ratio, we use the formula

r = a_{2}/a_{1}

General term of the geometric progression :

t_{n} = a r^{(n-1)}

**Example 1 :**

Find out which of the following sequences are geometric sequences . For those geometric sequences, find the common ratio.

(i) 0.12, 0.24, 0.48,.........

**Solution :**

t_{1} = 0.12, t_{2} = 0.24 and t_{3} = 0.48

r = t r = 0.24 / 0.12 r = 24 / 12 r = 2 ----(1) |
r = t r = 0.48 / 0.24 r = 48/24 r = 2 ----(2) |

Since the common ratios are same, the given sequence is geometric progression.

The required common ratio is 2.

(ii) 0.004, 0.02, 1, ..........

**Solution :**

**t _{1} = 0.004, t_{2} = 0.02 and t_{3} = 1**

r = t r = 0.02 / 0.004 r = 20 / 4 r = 5 ----(1) |
r = t r = 1 / 0.02 r = 100/2 r = 50 ----(2) |

Since the common ratios are not same, the given sequence is not a geometric progression.

The required common ratio is 5.

**Example 2 :**

Find the 10^{th} term and the common ratio of the geometric sequence

1/4, -1/2, 1, -2,............

**Solution :**

To find the 10^{th} terms of the G.P we use the formula given below.

t_{n} = a r^{(n-1)}

a = 1/4, r = (-1/2) / (1/4) ==> -2 and n = 10

t_{10} = (1/4) (-2)^{(10 - 1)}

t_{10} = (1/4) (-2)^{9}

t_{10} = (1/4) (-512)

t_{10} = -128

Therefore the 10^{th} and common ratio of the given geometric sequence are -128 and -2 respectively.

After having gone through the stuff given above, we hope that the students would have understood about geometric progression.

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