What is the sum to infinity of the given geometric progression?

2, 0.2, 0.02, 0.002, ......

Correct option is A

​$$\begin{aligned}&\textbf{Identify the First Term } (a) \text{ and Common Ratio } (r) \\[6pt]&\bullet\ \text{First term } (a):\ 2 \\[4pt]&\bullet\ \text{Common ratio } (r): \\&r = \frac{0.2}{2} = 0.1 \quad \text{(or)} \quad \frac{0.02}{0.2} = 0.1 \quad \text{(consistent)} \\[12pt]&\textbf{Check for Convergence} \\[4pt]&\text{For an infinite GP to converge (have a finite sum), the absolute value of the common ratio must satisfy:} \\&\qquad |r| < 1 \\[4pt]&\text{Here, } r = 0.1 \text{ meets this condition.} \\[12pt]&\textbf{Apply the Sum to Infinity Formula} \\[4pt]&\text{The sum } S_\infty \text{ of an infinite GP is given by:} \\&S_\infty = \frac{a}{1 - r} \\[6pt]&\text{Substitute } a = 2 \text{ and } r = 0.1: \\&S_\infty = \frac{2}{1 - 0.1} = \frac{2}{0.9} = \frac{20}{9}\end{aligned}$$​​