To find the final amount of the investment after 7 years with deposits made in the first and third years, we can create a polynomial model based on how each deposit grows over time with interest.
Let's break this down:
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Initial Deposit: $5,000 is deposited at the beginning of year 1. This deposit will accumulate interest for 7 years.
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Second Deposit: $2,000 is deposited at the beginning of year 3. This deposit will accumulate interest for 5 years (from the beginning of year 3 to the end of year 7).
Interest Growth Calculation
For the first deposit ($5,000 for 7 years):
The formula for the future value of a single deposit can be expressed as: \[ FV_1 = P_1 (1 + r)^n \] Where:
- \( P_1 = 5000 \) (the initial deposit)
- \( r \) is the interest rate
- \( n = 7 \) (number of years)
So, for the first deposit: \[ FV_1 = 5000 (1 + r)^7 \]
For the second deposit ($2,000 for 5 years):
Similarly: \[ FV_2 = P_2 (1 + r)^m \] Where:
- \( P_2 = 2000 \) (the second deposit)
- \( m = 5 \) (number of years from year 3 to year 7)
So, for the second deposit: \[ FV_2 = 2000 (1 + r)^5 \]
Total Future Value
Now, we can combine these two future values to get the total future value of the investment after 7 years: \[ C(x) = FV_1 + FV_2 \] Substituting in the values we calculated: \[ C(x) = 5000(1 + r)^7 + 2000(1 + r)^5 \]
Polynomial Representation
To write this in a polynomial form, we can factor out common terms if necessary, but typically, this is expressed as-is for clarity regarding the contribution of each deposit over time.
Thus, the polynomial \( C(x) \) representing the final amount after 7 years is: \[ C(r) = 5000(1 + r)^7 + 2000(1 + r)^5 \]
This polynomial can be evaluated for any specific interest rates by substituting the value of \( r \).
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