Let's break down the given expression to understand and solve it step by step. We need to evaluate the following expression:
[tex]\[ 300,000 \left[\frac{1 - (1 + 0.02)^{-12}}{0.02}\right] \][/tex]
1. Identify Constants and Variables:
- Principal ([tex]\(P\)[/tex]) = 300,000
- Interest rate ([tex]\(r\)[/tex]) = 0.02 (2%)
- Number of periods ([tex]\(n\)[/tex]) = 12
2. Evaluate the exponentiation term: [tex]\((1 + r)^{-n}\)[/tex]
- [tex]\( (1 + 0.02)^{-12} \)[/tex]
3. Calculate [tex]\((1 + 0.02)^{-12}\)[/tex]:
- [tex]\( = 1.02^{-12} \approx 0.7884931755816562 \)[/tex]
4. Subtract this value from 1:
- [tex]\(1 - 0.7884931755816562 = 0.2115068244183438\)[/tex]
5. Divide the result by the interest rate [tex]\(r\)[/tex]:
- [tex]\( \frac{0.2115068244183438}{0.02} \approx 10.57534122091719 \)[/tex]
6. Multiply this result by the principal [tex]\(P\)[/tex]:
- [tex]\( 300,000 \times 10.57534122091719 \approx 3,172,602.366275157 \)[/tex]
Thus, the fully calculated result is:
[tex]\[ 3,172,602.366275157 \][/tex]
And the term calculation before the final multiplication is [tex]\(0.7884931755816562\)[/tex]. Therefore, the final detailed step-by-step solution is:
- The term [tex]\((1 + 0.02)^{-12}\)[/tex] evaluates to approximately 0.7884931755816562.
- Plugging this back into the expression calculates the final result to be approximately 3,172,602.366275157.
