Introduction
While arranging numbers in order, sometimes, we might encounter that the ratio of any two consecutive numbers is always equal throughout the arrangement. In such cases, the ordered set of numbers is considered to be in a Geometric Progression, abbreviated as GP. It is a broad term that includes Geometric Sequence and Geometric Series within itself.
Example:
$$1,3,9,27,81$$
$$2,4,8,16,32$$
In each of the above-mentioned geometric progressions, the terms or numbers differ from their preceding by a constant ratio, called the common ratio of the progression and it is denoted by 'r'.
Mathematically, a progression is said to be a geometric progression if for each terms in the progression, there exists a common ratio, r which is defined as $$\text{r = } \dfrac{ \text{t}_\text{n}}{ \text{t}_\text{n-1}}$$.
General Term of a Geometric Progression
From the concept of common ratio, we can consider the following arrangement to be in geometric progression provided that a and r are numbers.
$$\text{a},\text{ar},\text{ar}^2,\text{ar}^3,...$$
Let us assume that the above progression has n terms then, it should be noted that the nth term in the progression has an additional term equal to the product of the first term and the common ratio raised to the power of (n-1).
Hence, it can be concluded that the nth term of the geometric progression can be denoted by $$\text{ar}^\text{n-1}$$ which is also the formula for the last term of the progression.
Properties of Geometric Progression
Let us consider a geometric progression as follows:
$$\text{a}_0,\text{a}_1,\text{a}_2,\text{a}_3,...$$
Let k $\in$ R and k $\neq$ 0 then numbers in the above progression obey the following properties:
- If each term is multiplied or divided by k then the new progression is again in Geometric Progression.
$$\text{ka}_0,\text{ka}_1,\text{ka}_2,\text{ka}_3,...$$
$$\dfrac{\text{a}_0}{\text{k}},\dfrac{\text{a}_1}{\text{k}},\dfrac{\text{a}_2}{\text{k}},\dfrac{\text{a}_3}{\text{k}},...$$ - If each term in the GP is inversed, then the resulting progression is also in Geometric Progression.
$$\dfrac{1}{\text{a}_0},\dfrac{1}{\text{a}_1},\dfrac{1}{\text{a}_2},\dfrac{1}{\text{a}_3},...$$ - If each term in the GP is raised to the same power, then the resulting progression is again in GP.
$$\text{a}_0^k,\text{a}_1^k,\text{a}_2^k,\text{a}_3^k,...$$