Difficulty: Easy
Correct Answer: All of these
Explanation:
Introduction / Context:Many engineering and business cash flows do not remain constant; they change by a fixed amount every period—think of scheduled maintenance ramp-ups or planned annual reductions. Such patterns are modeled using arithmetic (uniform) gradients, which are indispensable in present-worth and annual-worth analyses.
Given Data / Assumptions:
Concept / Approach:An arithmetic gradient adds a constant amount G each period to the base annuity; geometric gradients instead grow by a constant percentage. The sign of G establishes whether the series is rising (positive gradient) or falling (negative gradient). Correct identification of the gradient type is crucial to applying the right factor formulas.
Step-by-Step Solution:
Define gradient: difference between successive cash-flow amounts.Uniform (arithmetic) gradient: difference equals constant G each period.Assign sign: G > 0 → positive gradient; G < 0 → negative gradient.Verification / Alternative check:
Model an example: A base annuity A with gradient G for n periods; convert series to present worth using gradient factor P/G at interest rate i.Why Other Options Are Wrong:
Each of a–c is correct; “All of these” is the comprehensive answer.“None of these” conflicts with standard gradient definitions.Common Pitfalls:
Confusing arithmetic gradients with geometric escalation; use the correct factor (P/G versus P/A with growth).Forgetting that gradients typically start in period 2 by convention in factor notation.Final Answer:
All of these
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