Question
Solve Ai equals Upper R left bracket 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Baseline right bracket for A.
Question content area bottom
Part 1
Choose the correct choice below.
A.Upper A equals Ri left bracket 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Baseline right bracket
Upper A equals Ri left bracket 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Baseline right bracket
B.Upper A equals Upper R left parenthesis StartFraction 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Over i EndFraction right parenthesis
Upper A equals Upper R left parenthesis StartFraction 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Over i EndFraction right parenthesis
C.Upper A equals Ri left parenthesis left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 right parenthesis
Upper A equals Ri left parenthesis left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 right parenthesis
D.Upper A equals Upper R left parenthesis StartFraction left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 Over t EndFraction right parenthesis
Upper A equals Upper R left parenthesis StartFraction left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 Over t EndFraction right parenthesis
Question content area bottom
Part 1
Choose the correct choice below.
A.Upper A equals Ri left bracket 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Baseline right bracket
Upper A equals Ri left bracket 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Baseline right bracket
B.Upper A equals Upper R left parenthesis StartFraction 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Over i EndFraction right parenthesis
Upper A equals Upper R left parenthesis StartFraction 1 minus left parenthesis 1 plus i right parenthesis Superscript negative n Over i EndFraction right parenthesis
C.Upper A equals Ri left parenthesis left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 right parenthesis
Upper A equals Ri left parenthesis left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 right parenthesis
D.Upper A equals Upper R left parenthesis StartFraction left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 Over t EndFraction right parenthesis
Upper A equals Upper R left parenthesis StartFraction left parenthesis 1 plus i right parenthesis Superscript negative n Baseline minus 1 Over t EndFraction right parenthesis
Answers
GPT-4o mini
To solve the equation \( A_i = R \left( 1 - (1 + i)^{-n} \right) \) for \( A \), we first need to isolate \( A \).
The equation as it is only relates \( A_i \) (which should actually be \( A \) to match the given context). Thus, we can rewrite the equation as:
\[ A = R \left( 1 - (1 + i)^{-n} \right) \]
Now let’s compare this expression to the answer choices given:
1. A. \( A = R i \left( 1 - (1 + i)^{-n} \right) \)
2. B. \( A = R \left( \frac{1 - (1 + i)^{-n}}{i} \right) \)
3. C. \( A = R i \left( (1 + i)^{-n} - 1 \right) \)
4. D. \( A = R \left( \frac{(1 + i)^{-n} - 1}{t} \right) \)
From the expression we derived, it matches option B:
### **B. \( A = R \left( \frac{1 - (1 + i)^{-n}}{i} \right) \)**
To verify: If you multiply the right-hand side of option B by \( i \):
\[ A = R \cdot i \left( \frac{(1 - (1 + i)^{-n})}{i} \right) = R \left( 1 - (1 + i)^{-n} \right) \]
So the correct answer is:
**B. \( A = R \left( \frac{1 - (1 + i)^{-n}}{i} \right) \)**
The equation as it is only relates \( A_i \) (which should actually be \( A \) to match the given context). Thus, we can rewrite the equation as:
\[ A = R \left( 1 - (1 + i)^{-n} \right) \]
Now let’s compare this expression to the answer choices given:
1. A. \( A = R i \left( 1 - (1 + i)^{-n} \right) \)
2. B. \( A = R \left( \frac{1 - (1 + i)^{-n}}{i} \right) \)
3. C. \( A = R i \left( (1 + i)^{-n} - 1 \right) \)
4. D. \( A = R \left( \frac{(1 + i)^{-n} - 1}{t} \right) \)
From the expression we derived, it matches option B:
### **B. \( A = R \left( \frac{1 - (1 + i)^{-n}}{i} \right) \)**
To verify: If you multiply the right-hand side of option B by \( i \):
\[ A = R \cdot i \left( \frac{(1 - (1 + i)^{-n})}{i} \right) = R \left( 1 - (1 + i)^{-n} \right) \]
So the correct answer is:
**B. \( A = R \left( \frac{1 - (1 + i)^{-n}}{i} \right) \)**