To figure out the final amounts for each investment over a 7-year term at an interest rate of 3.7%, we start by converting the interest rate into the variable \( x = 1 + r \). Here, \( r = 0.037 \) (which is 3.7% as a decimal). So, we have:
\[ x = 1 + 0.037 = 1.037 \]
Investment 1
For Investment 1, you deposit $4,000 at the beginning of the first year. This amount will earn interest for all 7 years. We can use the formula for compound interest to model this investment:
\[ C_1(x) = 4000 \cdot x^7 \]
Substituting \( x = 1.037 \):
\[ C_1(1.037) = 4000 \cdot (1.037)^7 \]
Calculating \( (1.037)^7 \):
\[ (1.037)^7 \approx 1.287 \]
Now multiply by $4,000:
\[ C_1(1.037) \approx 4000 \cdot 1.287 \approx 5148 \]
Investment 2
For Investment 2, the deposits are $2,000 at the beginning of the first year, $1,500 at the beginning of the third year, and $500 at the beginning of the fifth year. We need to calculate the final amount for each of these deposits separately, taking into account how many years each deposit earns interest.
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First deposit ($2,000 at year 1): This earns interest for 7 years: \[ C_{2,1}(x) = 2000 \cdot x^7 \] \[ C_{2,1}(1.037) = 2000 \cdot (1.037)^7 \approx 2000 \cdot 1.287 \approx 2574 \]
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Second deposit ($1,500 at year 3): This earns interest for 5 years: \[ C_{2,2}(x) = 1500 \cdot x^5 \] \[ C_{2,2}(1.037) = 1500 \cdot (1.037)^5 \approx 1500 \cdot 1.199 \approx 1799 \]
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Third deposit ($500 at year 5): This earns interest for 3 years: \[ C_{2,3}(x) = 500 \cdot x^3 \] \[ C_{2,3}(1.037) = 500 \cdot (1.037)^3 \approx 500 \cdot 1.114 \approx 557 \]
Now, we sum all these amounts for Investment 2:
\[ C_2(1.037) \approx 2574 + 1799 + 557 \approx 4930 \]
Comparing Investments
- Investment 1: Approximately $5,148
- Investment 2: Approximately $4,930
Conclusion
Investment 1 results in more total money at the end of the 7 years.
Reasoning on Future Investments
When comparing both options, investing more at one time (like in Investment 1) can lead to a higher return, because the money is earning interest for a longer period. In Investment 2, the smaller deposits don't earn interest as long, which means that you may end up earning less overall compared to making a larger single investment at the beginning.
In general, if you have the option to invest a lump sum, it's usually better to do so, as it maximizes the amount of time your money can grow through interest.