A closed-loop inverting amplifier has a gain magnitude of 3.9. If the input resistor Rin = 1.6 kΩ, what value of feedback resistor Rf is required to realize this gain?

Introduction:
Inverting op-amp gain design is a straightforward resistor ratio problem. This item confirms fluency with the formula and with consistent unit handling between ohms and kilohms.

Given Data / Assumptions:

• Desired closed-loop magnitude |Av| = 3.9.
• Rin = 1.6 kΩ.
• Ideal op-amp approximation for gain computation.

Concept / Approach:
For an inverting amplifier, the magnitude of the closed-loop gain is |Av| = Rf / Rin. Solve for Rf by multiplying the target gain by Rin.

Step-by-Step Solution:
1) Use |Av| = Rf / Rin.2) Rearrange: Rf = |Av| * Rin.3) Substitute: Rf = 3.9 * 1.6 kΩ = 6.24 kΩ.4) Convert to ohms if desired: 6.24 kΩ = 6240 Ω.

Verification / Alternative check:
Check back: |Av| = 6240 Ω / 1600 Ω = 3.9, confirming the design value.

Why Other Options Are Wrong:
2.4 kilohms and 0.62 kilohms: These produce gains of 1.5 and 0.39 respectively, not 3.9.

410 ohms: Yields a gain magnitude of approximately 0.256, far from the target.

6.24 megaohms: Off by 1000×; unit mistake between kΩ and MΩ.

Common Pitfalls:
Mixing units (kΩ vs Ω) or overlooking that the sign of gain is negative in inverting mode, while the question asks only for magnitude.

Final Answer:
6240 ohms