For compound interest at rate i per period on principal P over n periods, which expression gives the total compounded interest accrued (i.e., amount earned above principal)?

Introduction / Context:
Capital budgeting and plant economics frequently require computing the growth of money under compound interest. Distinguishing between the future amount (principal plus interest) and the interest earned alone is essential for cash-flow analysis and depreciation planning.

Given Data / Assumptions:

• Principal = P at time zero.
• Interest rate per period = i.
• Number of compounding periods = n.

Concept / Approach:
The compound amount formula is F = P * (1 + i)^n. The compounded interest (earned amount above principal) equals F - P. Therefore, interest accrued = P * ((1 + i)^n - 1). The other expressions either give the total future value or are incorrect linear/exponential forms for compound growth.

Step-by-Step Solution:
Start with F = P * (1 + i)^n.Interest accrued, I_c = F - P.I_c = P * (1 + i)^n - P = P * ((1 + i)^n - 1).

Verification / Alternative check:
For i = 0, the expression gives zero interest: P * ((1 + 0)^n - 1) = 0, which is consistent. For one period (n = 1), I_c = P * ((1 + i) - 1) = P * i, matching simple interest for a single period.

Why Other Options Are Wrong:
P * (1 + i)^n: This is the future amount, not the interest alone.P * (1 - i)^n: Represents decay, not compound growth at positive i.P * (1 + i * n): Linear growth corresponds to simple interest, not compounding.

Common Pitfalls:

• Confusing future amount with interest portion.
• Using simple interest formula for multi-period compounding.
• Forgetting to convert nominal to effective rates when compounding frequency changes.

Final Answer:
P * ((1 + i)^n - 1)