The first term of an arithmetic progression is 2. Its second and sixth terms are consecutive terms of a geometric progression. Find the common difference if the arithmetic progression is increasing.

Let the first term of the AP be $a$ and the common difference be $d$.
Given that the first term of the AP is 2, so $a=2$.
The terms of the AP are:
First term: $T_1 = a = 2$
Second term: $T_2 = a+d = 2+d$
Sixth term: $T_6 = a+5d = 2+5d$

The first, second, and sixth terms of the AP are consecutive terms of a geometric progression.
These terms are 2, $2+d$, and $2+5d$.
In a geometric progression, the ratio of consecutive terms is constant.
Therefore, we can write:

Cross-multiply the terms:

Expand the left side:

Rearrange the terms to form a quadratic equation:

Factor out $d$:

This gives two possible values for $d$:

The problem states that it is an increasing arithmetic progression.
If $d=0$, the AP terms would be $2, 2, 2, \dots$, which is not an increasing progression.
If $d=6$, the AP terms would be $2, 2+6, 2+2(6), \dots$, which are $2, 8, 14, \dots$. This is an increasing arithmetic progression.
Therefore, the common difference must be 6.