To model the situation mathematically, we can interpret the grandfather's contributions to the bank account and the total balance over the years.
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Initial Contribution:
- At the first birthday (Year 0), the grandfather deposits $100.
- So, the balance after the first birthday (Year 0) is: \[ B(0) = 100 \]
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Subsequent Contributions:
- At each subsequent birthday (Year 1 and beyond), the grandfather deposits $80.
- For Year 1, the total balance becomes: \[ B(1) = B(0) + 80 = 100 + 80 = 180 \]
- For Year 2: \[ B(2) = B(1) + 80 = 180 + 80 = 260 \]
- For Year 3: \[ B(3) = B(2) + 80 = 260 + 80 = 340 \]
So far, we have:
- Year 0: Balance = 100
- Year 1: Balance = 180
- Year 2: Balance = 260
- Year 3: Balance = 340
- General Function: From the above, we can see that the balance increases by $80 each year after the first year. Thus, we can generalize the balance function for any year \( n \):
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For Year 0 (when the voice first birthday occurs): \[ B(0) = 100 \]
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For Year 1 (when the first deposit occurs): \[ B(1) = 100 + 80(1) \]
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For Year \( n \) (where \( n \geq 0 \)): \[ B(n) = 100 + 80n \]
Thus, we can express the function that models the amount of money in the bank account at any year \( n \) as: \[ B(n) = 100 + 80n \]
Where \( n \) is the number of years after the first birthday (i.e., \( n = 0 \) for the first birthday, \( n = 1 \) for the second birthday, and so on).