Let's simplify the two expressions:

(12:17)^x:2 + 1 = (5:20)^x:2 + 1

(12/17)^x/2 + 1 = (1/4)^x/2 + 1

(12/17)^x/2 + 1 = (1/2)^x + 1

Now, we need to solve for x. Let's set up the equation:

(12/17)^x/2 + 1 = (1/2)^x + 1

(12/17)^x/2 = (1/2)^x

Taking the natural logarithm of both sides:

ln((12/17)^x/2) = ln((1/2)^x)

Simplifying further:

(x/2) ln(12/17) = x ln(1/2)

(x/2) ln(12) - (x/2) ln(17) = x * (-ln(2))

(x/2) [ln(12) - ln(17)] = -x ln(2)

x/2 = -x * ln(2) / [ln(12) - ln(17)]

x = 2 [-x ln(2) / [ln(12) - ln(17)]]

x = -2 [x ln(2) / [ln(12) - ln(17)]]

Since these equations cannot be simplified further without a numerical value, the value of x will depend on the specific value of x.