Identify an Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference ($d$). Let's check each option: (A) $2, 4, 8, 16, ......$ Differences: $4-2=2$, $8-4=4$. The differences are not constant, so it's not an AP (it's a Geometric Progression).

(B) $3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, ......$ Differences: $(3 + \sqrt{2}) - 3 = \sqrt{2}$ $(3 + 2\sqrt{2}) - (3 + \sqrt{2}) = 3 + 2\sqrt{2} - 3 - \sqrt{2} = \sqrt{2}$ The common difference is constant ($\sqrt{2}$), so this is an AP.

(C) $1, 3, 9, 27, ......$ Differences: $3-1=2$, $9-3=6$. The differences are not constant, so it's not an AP (it's a Geometric Progression).

(D) $\sqrt{3}, \sqrt{6}, \sqrt{9}, ......$ Differences: $\sqrt{6} - \sqrt{3} = \sqrt{3}(\sqrt{2} - 1)$ $\sqrt{9} - \sqrt{6} = 3 - \sqrt{6}$ The differences are not constant, so it's not an AP.