To find the value of A, we need to evaluate the sum of the products of three consecutive numbers from 3 to 22.

Let's break down the expression:

A = 3.4.5 + 4.5.6 + 5.6.7 + .... + 22.23.24

We can rewrite this expression as:

A = (3*4*5) + (4*5*6) + (5*6*7) + .... + (22*23*24)

Now, let's simplify each term:

A = 60 + 120 + 210 + .... + (22*23*24)

To find the sum of these terms, we can use the formula for the sum of an arithmetic series:

Sn = (n/2)(a1 + an)

where Sn is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term.

In this case, the first term a1 is 60, the last term an is (22*23*24), and the number of terms n is 22 - 3 + 1 = 20.

Plugging these values into the formula, we get:

A = (20/2)(60 + (22*23*24))

A = 10(60 + 12,144)

A = 10(12,204)

A = 122,040

Therefore, the value of A is 122,040.