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Sagot :
To find the periodic payment that will amount to a future value of [tex]$21,000 when an interest rate of 6% (compounded annually) is applied over 12 consecutive years, we can proceed as follows:
### Step-by-Step Solution:
1. Understand the Terms and the Formula:
- Future Value (FV) = $[/tex]21,000
- Interest Rate (r) = 6% or 0.06 (as a decimal)
- Number of Periods (n) = 12 years
We will use the formula for the future value of an ordinary annuity:
[tex]\[ FV = PMT \times \left[\frac{{(1 + r)^n - 1}}{r}\right] \][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value.
- [tex]\( PMT \)[/tex] is the periodic payment.
- [tex]\( r \)[/tex] is the interest rate per period.
- [tex]\( n \)[/tex] is the number of periods.
2. Rearrange the Formula to Solve for [tex]\( PMT \)[/tex]:
We need to solve for the periodic payment [tex]\( PMT \)[/tex]:
[tex]\[ PMT = \frac{FV}{\left[\frac{{(1 + r)^n - 1}}{r}\right]} \][/tex]
3. Substitute the Known Values:
- [tex]\( FV = 21000 \)[/tex]
- [tex]\( r = 0.06 \)[/tex]
- [tex]\( n = 12 \)[/tex]
So, the rearranged formula becomes:
[tex]\[ PMT = \frac{21000}{\left[\frac{{(1 + 0.06)^{12} - 1}}{0.06}\right]} \][/tex]
4. Calculate the Denominator:
Let's break this down:
- Calculate [tex]\( (1 + 0.06)^{12} \)[/tex]:
[tex]\[ (1 + 0.06)^{12} = 1.06^{12} \][/tex]
- Subtract 1 from this result:
[tex]\[ 1.06^{12} - 1 \][/tex]
- Divide by the interest rate [tex]\( r = 0.06 \)[/tex]:
[tex]\[ \frac{1.06^{12} - 1}{0.06} \][/tex]
5. Complete the Calculation:
After calculating the entire expression in the denominator, plug back into the formula:
[tex]\[ PMT = \frac{21000}{\left[\frac{(1 + 0.06)^{12} - 1}{0.06}\right]} \][/tex]
6. Round the Result:
Ensure the final answer is rounded to the nearest cent.
The periodic payment will amount to approximately $1244.82.
- Interest Rate (r) = 6% or 0.06 (as a decimal)
- Number of Periods (n) = 12 years
We will use the formula for the future value of an ordinary annuity:
[tex]\[ FV = PMT \times \left[\frac{{(1 + r)^n - 1}}{r}\right] \][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value.
- [tex]\( PMT \)[/tex] is the periodic payment.
- [tex]\( r \)[/tex] is the interest rate per period.
- [tex]\( n \)[/tex] is the number of periods.
2. Rearrange the Formula to Solve for [tex]\( PMT \)[/tex]:
We need to solve for the periodic payment [tex]\( PMT \)[/tex]:
[tex]\[ PMT = \frac{FV}{\left[\frac{{(1 + r)^n - 1}}{r}\right]} \][/tex]
3. Substitute the Known Values:
- [tex]\( FV = 21000 \)[/tex]
- [tex]\( r = 0.06 \)[/tex]
- [tex]\( n = 12 \)[/tex]
So, the rearranged formula becomes:
[tex]\[ PMT = \frac{21000}{\left[\frac{{(1 + 0.06)^{12} - 1}}{0.06}\right]} \][/tex]
4. Calculate the Denominator:
Let's break this down:
- Calculate [tex]\( (1 + 0.06)^{12} \)[/tex]:
[tex]\[ (1 + 0.06)^{12} = 1.06^{12} \][/tex]
- Subtract 1 from this result:
[tex]\[ 1.06^{12} - 1 \][/tex]
- Divide by the interest rate [tex]\( r = 0.06 \)[/tex]:
[tex]\[ \frac{1.06^{12} - 1}{0.06} \][/tex]
5. Complete the Calculation:
After calculating the entire expression in the denominator, plug back into the formula:
[tex]\[ PMT = \frac{21000}{\left[\frac{(1 + 0.06)^{12} - 1}{0.06}\right]} \][/tex]
6. Round the Result:
Ensure the final answer is rounded to the nearest cent.
The periodic payment will amount to approximately $1244.82.
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