If $2500 is placed in an account which pays 7% per annum, compounding annually, then the balance in the account after n years is calculated by:单项选择题
A
A. $2500 × 0.7n
B
B. $2500 × 7n
C
C. $175n
D
D. $2500 × 1.07n

Question

If $2500 is placed in an account which pays 7% per annum, compounding annually, then the balance in the account after n years is calculated by:

BlackTom AI · Accepted Answer

The compound interest formula is A = P(1 + r)^n, so use P = 2500 and r = 0.07.

1. Relevant knowledge points and concepts
- Principal P is the initial deposit ($2500).
- Annual nominal interest rate r = 7% = 0.07 (expressed as a decimal).
- Compounding annually means the amount grows by a factor (1 + r) each year.
- After n years the balance A = P(1 + r)^n.

2. Complete reasoning process step by step
- Start with the principal: P = $2500.
- Each year the balance is multiplied by (1 + r) = 1 + 0.07 = 1.07.
- After n years the repeated multiplication gives A = 2500 × (1.07)^n.

3. Calculations and intermediate steps
- Write the formula: A = 2500(1 + 0.07)^n.
- Simplify inside parentheses: 1 + 0.07 = 1.07.
- Final expression: A = 2500 × 1.07^n.
- Example checks: n = 1 → A = 2500 × 1.07 = $2675; n = 2 → A = 2500 × 1.07^2 = 2500 × 1.1449 = $2862.25.

4. Why the correct answer is correct
- Compounding multiplies the entire balance each year by 1.07, which is precisely represented by the exponent n in 1.07^n. The expression 2500 × 1.07^n captures both the initial principal and the repeated growth.

5. Why the other options are incorrect
- A: 2500 × 0.7^n implies a 30% annual decay (multiply by 0.7), which is the opposite of growth.
- B: 2500 × 7n treats the rate as 7 times n (linear scaling) and gives nonsensical units; it is not compounding.
- C: 175n equals 7% of 2500 times n, which is simple interest (no compounding). For n = 2 this gives 2500 + 350 = 2850, different from compound result 2862.25.

6. Conclusion
- The correct formula is A = 2500 × 1.07^n, matching option D.

If $2500 is placed in an account which pays 7% per annum, compounding annually, then the balance in the account after n years is calculated by:单项选择题

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