To find the value of A, we can use the formula for the sum of a geometric series:

S = a * (r^n - 1) / (r - 1)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

In this case, a = 2, r = 2, and n = 100. Plugging these values into the formula, we get:

A = 2 * (2^100 - 1) / (2 - 1)
A = 2 * (2^100 - 1)
A = 2^101 - 2

To find the value of B, we can use the same formula with a = 1 and r = 3:

B = 1 * (3^101 - 1) / (3 - 1)
B = 3^101 - 1

Finally, to find the value of C, we can use the formula for the sum of a geometric series again, but with a = -2 and r = -2:

C = -2 * (-2^101 - 1) / (-2 - 1)
C = -2 * (-2^101 - 1)
C = 2^102 + 2

So the values of A, B, and C are:

A = 2^101 - 2
B = 3^101 - 1
C = 2^102 + 2