**2. Geometric progression**

**(A) Characteristics of Geometric Progressions**

Geometric progression is a progression in which the ratio of any term to the immediate term before is a constant. The constant is called common ratio,

*r.*

**Example:**

Determine whether or not each of the following number sequences is a geometric progression (GP).

**(a)**1, 4, 16, 64, …..

**(b)**10, –5, 2.5, –1.25, …..

**(c)**2, 4, 12, 48, …..

**(d)**–6, 1, 8, 15, …..

Solution:Solution:

*[Smart Tips: For GP, times a fixed number every time to get the next time.]***(a)**

**This is a GP,**

*a*=1,*r*= 4.

**(b)**

$\begin{array}{l}\text{Common ratio,}\\ r=\frac{{T}_{2}}{{T}_{1}}=-\frac{5}{10}=-\frac{1}{2}\end{array}$

**This is a GP,**

*a*=1,*r*= –**½**

**.**

**(c)**

**This is NOT a GP.**

This is because the ratio of each term to its preceding term is not a similar constant.

**(B) The steps to prove whether a given number sequence is a geometric progression.**

**Step 1**: List down any three consecutive terms. [Example:

*T*

_{1},

*T*

_{2},

*T*

_{3}.]

**Step 2**: Calculate the values of $\frac{{T}_{3}}{{T}_{2}}\text{and}\frac{{T}_{2}}{{T}_{1}}.$

**Step 3**: If $\frac{{T}_{3}}{{T}_{2}}=\frac{{T}_{2}}{{T}_{1}}=r$ , then the number sequence is a geometric

progression.

**Step 4**: If $\frac{{T}_{3}}{{T}_{2}}\ne \frac{{T}_{2}}{{T}_{1}}$ , then the number sequence is not a geometric progression.