Phil bought a car by taking out a loan for 18,500 at 0.5 %...

Phil bought a car by taking out a loan for $$\$ 18,500$$ at $0.5 \%$ interest per month. Phil's normal monthly payment is $$\$ 434.47$$ per month, but he decides that he can afford to pay $$\$ 100$$ extra toward the balance each month. His balance each month is given by the recursively defined sequence $$ B_0=18,500 \quad B_n=1.005 B_{n-1}-534.47 $$ Determine Phil's balance after making the first payment. That is, determine $B_1$.

Phil bought a car by taking out a loan for $$\$ 18,500$$ at $0.5 \%$ interest per month. Phil's normal monthly payment is $$\$ 434.47$$ per month, but he decides that he can afford to pay $$\$ 100$$ extra toward the balance each month. His balance each month is given by the recursively defined sequence
$$
B_0=18,500 \quad B_n=1.005 B_{n-1}-534.47
$$
Determine Phil's balance after making the first payment. That is, determine $B_1$.

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Key Concepts

Compound Interest

Compound interest is the process by which interest is added to the principal, so that from that moment on, the interest that has been added also earns interest. In financial contexts, this means that each period's interest is calculated on the sum of the initial principal and all accumulated interest from previous periods. This concept is fundamental when calculating the growth of debt or investment over time, as shown by the monthly multiplication factor in the recurrence.

Recurrence Relations

A recurrence relation is a way of defining a sequence where each term is formulated as a function of its preceding terms. In the context of finance, recursive formulas are often used to model how a loan balance changes over time with interest and regular payments. The current balance is derived from the previous balance by applying the interest factor and then subtracting the payment amount.

Loan Amortization

Loan amortization refers to the systematic process of paying off a debt over time through scheduled, periodic payments that cover both the interest and a portion of the principal balance. Extra payments can shorten the repayment period and reduce the overall interest paid. This concept is illustrated in the problem by showing how the payment exceeds the standard amount, affecting the balance calculation in the recurrence.

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