[THOUGHT] We need to compute the contour integral of f(z) = z^2 over the upper semicircular arc C: z = (5/2) e^{iθ}, θ from 0 to π. But careful: The problem says "C is traversed from z = 5/2 to z = -5/2". That corresponds to θ going from 0 to π? Let's check: z = (5/2) e^{iθ}. At θ=0, z = 5/2. At θ=π, z = (5/2) e^{iπ} = -5/2. Yes, so that's the upper semicircle (since θ from 0 to π goes through upper half-plane). So C is the upper semicircle radius 5/2.

We need to compute ∫_C z^2 dz. Parametrize: z(θ) = (5/2) e^{iθ}, dz = (5/2) i e^{iθ} dθ. Then z^2 = (25/4) e^{2iθ}. So integrand: z^2 dz = (25/4) e^{2iθ} * (5/2) i e^{iθ} dθ = (25/4)*(5/2)*i e^{3iθ} dθ = (125/8) i e^{3iθ} dθ.

Thus ∫C z^2 dz = ∫{θ=0}^{π} (125/8) i e^{3iθ} dθ.

Compute integral: ∫_0^π e^{3iθ} dθ = [ e^{3iθ}/(3i) ]_0^π = (e^{3iπ} - 1)/(3i). e^{3iπ} = e^{iπ} cubed? Actually e^{3iπ} = cos(3π) + i sin(3π) = -1 + 0i = -1. So numerator: -1 - 1 = -2. So integral = (-2)/(3i) = multiply numerator and denominator by i: (-2 i)/(3 i^2) = (-2 i)/(-3) = (2 i)/3? Wait: (-2)/(3i) = (-2)(1/(3i)) = (-2)(i/(3i^2))? Actually 1/i = -i. So (-2)/(3i) = (-2/3)(1/i) = (-2/3)(-i) = (2i)/3. Yes.

Thus ∫_0^π e^{3iθ} dθ = (2i)/3.

Multiply by (125/8) i: (125/8) i * (2i)/3 = (125/8)(2i^2)/3 = (125/8)(2*(-1))/3 = (125/8)*(-2)/3 = (-250)/(24) = simplify: divide numerator and denominator by 2: -125/12.

So ∫_C f(z) dz = -125/12.

But the problem says ∫_C f(z) dz = -p/q, where p,q positive integers relatively prime. So -p/q = -125/12, thus p/q = 125/12. Since 125 and 12 are coprime (125=5^3, 12=2^2*3, no common factors), p=125, q=12. Then p+q = 125+12 = 137.

But wait: The problem statement says "Suppose the contour integral ∫_C f(z) dz = -p/q". So indeed we have -125/12, so p=125, q=12, sum=137.

However, I should verify using Python to avoid any algebra mistakes.

I'll compute using Python with symbolic integration or numeric integration.

Let's code. [/PYTHON] [PYTHON] import cmath

Parametrization: z(theta) = (5/2)exp(itheta), theta from 0 to pi

integrand: z^2 dz = ( (5/2)exp(itheta) )^2 * (5/2)iexp(i*theta) dtheta

= (25/4)exp(2itheta) * (5/2)iexp(itheta) dtheta

= (125/8)iexp(3itheta) dtheta

Compute integral analytically:

∫ (125/8)iexp(3itheta) dtheta from 0 to pi = (125/8)i * [exp(3itheta)/(3i)] from 0 to pi

= (125/8)i * (1/(3i)) * (exp(3ipi) - 1) = (125/8)(1/3)(exp(3ipi) - 1)

= (125/24)(exp(3i*pi) - 1)

exp(3ipi) = cos(3π) + i sin(3π) = -1

value = (125/24)*(-1 - 1) print("Exact value:", value)

Alternatively numeric integration:

import numpy as np

def integrand(theta): z = 2.5 * np.exp(1j * theta) dz = 2.5 * 1j * np.exp(1j * theta) return z**2 * dz

Use trapezoidal rule for integration

theta = np.linspace(0, np.pi, 100001) integrand_vals = integrand(theta) integral = np.trapz(integrand_vals, theta) print("Numerical integral:", integral) print("Real part:", integral.real) print("Imag part:", integral.imag) [/PYTHON]

[THOUGHT] The exact value computed is -125/12 = -10.416666..., matching the Python output. So p = 125, q = 12, p+q = 137.

$\boxed{137}$