91Ó°ÊÓ

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

For example, thesequence2, 6, 18, 54, ... is ageometric progressionwith common ratio 3.

The ration of 2^nd term and first term is called the common ratio.

The sequence:$2,\text{ â¶Ä‰}−6\text{ â¶Ä‰},\text{ â¶Ä‰}18\text{ â¶Ä‰}\mathrm{.....}$

Each term in a geometric sequence can be expressed in terms of the first term$a_1$and the common ratio r. Since each succeeding term is formulated from one or more previous terms, this is a recursive formula.

First term $a_1=2$and the common difference is:

$$\begin{gathered} r=\frac{a_2}{a_1} \\ =\frac{−6}{2} \\ =−3 \end{gathered}$$

In order to calculate the next three terms of the geometric sequence substitute$n=4,5,6$ and 2 for$a_1$ into the formula $a_n=a_1{r}^{n−1}$.

Thus, next three terms of the sequence$2,\text{ â¶Ä‰}−6\text{ â¶Ä‰},\text{ â¶Ä‰}18\text{ â¶Ä‰}\mathrm{.....}$are$−54,\text{ â¶Ä‰}162,\text{ â¶Ä‰}−486$.