# Geometric Sequence or Geometric Progression

__Introduction__:

Geometric sequence is a second type of sequence. One important application of geometric progression is in computing interest on saving account. Other applications can be found in biology and physics. Geometric sequence has also an application for finding the national income or population for any particular future year provided the values for the current year and the growth rate.

__Geometric Sequence__:

A **geometric** (**exponential**) **sequence** or **progression** (abbreviated as **G.P**) is sequence of numbers in which each term, after first, is obtained by multiplying the preceding term by a fixed number. The **fixed number** is called **common ratio **and is usually denoted by .

To determine whether or not a sequence of numbers is a G.P., we divide each member by the one which precedes it.

For example, the sequence

- is a G.P with
- is a G.P with
- is a G.P with

In other words, quantities are said to be in Geometric Progression (Abbreviated as G.P) when they increase or decrease by a **constant factor**.

__The nth term of a Geometric Sequence__:

Let be the first term and be the common ratio. Then, the second term is ; the third term is . In each of these terms the exponent of is less than the number of the term. Similarly, the term is the after the first and is found by multiplying by factors , or by . Hence, if represents the term, then

__Example__:

Given that are the first three terms of a geometric sequence, find the fifth term of the sequence.

__Solution__:

Here ,

By using the formula,

The fifth term,

__Example__:

What is the first term of a six-term geometric sequence in which the ratio is and the sixth term is ?

__Solution__:

Here , ,

To find by the formula