Definition of Geometric Progression (G.P)

What is Geometric Progression?

A sequence in which each term is obtained from the preceding term by multiplying or dividing by a constant factor is called the geometric progression or G.P.

A geometric sequence is a sequence such that any element after the first term is obtained by multiplying or dividing the preceding element by a constant factor called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

\( \scriptsize r = \normalsize \frac{a_2}{a_1} = \frac{a_3}{a_2} = ….\frac{a_n}{a_{n\:-\:1}} \)

The geometric sequence is also known as the Geometric Progression or G.P.

For example, the sequence 2, 4, 8, 16, 32 is a geometric sequence. You can notice that after the first term, the next term is obtained by multiplying the preceding element by 2. Here we say the common ration is 2.

To find the nth term of a geometric sequence we use the formula:

\( \scriptsize T_n = ar^{n\:-\:1} \)

Example 3.1.1:

A G.P is given as logx², logx⁶, logx¹⁸ ,… What is the common ratio?

Solution

Recall

logx² = 2logx

logx⁶ = 6logx

logx¹⁸ = 18logx

common ratio r = \( \frac{6logx}{2logx} \scriptsize = 3\)

i.e. common ratio = 3

Example 3.1.2:

In a G.P, the second and the fourth terms are 0.04 and 1 respectively.

Find the

i. Common ratio
ii. First ratio

Solution