A bank account earns 5% interest compounded monthly. Suppose...

A bank account earns 5$\%$ interest compounded monthly. Suppose that $\$ 1000$ is initially deposited into the account, but that $\$ 10$ is withdrawn each month. a. Show that the amount in the account after $n$ months is $A_{n}=(1+.05 / 12) A_{n-1}-10$ ; $A_{0}=1000.$ b. How much money will be in the account after 1 year? c. Is the amount increasing or decreasing? d. Suppose that instead of $\$ 10,$ a fixed amount $d$ dollars is withdrawn each month. Find a value of $d$ such that the amount in the account after each month remains $\$ 1000$ . e. What happens if $d$ is greater than this amount?

A bank account earns 5$\%$ interest compounded monthly. Suppose that $\$ 1000$ is initially deposited into the account, but that $\$ 10$ is withdrawn each month.
a. Show that the amount in the account after $n$ months is $A_{n}=(1+.05 / 12) A_{n-1}-10$ ; $A_{0}=1000.$
b. How much money will be in the account after 1 year?
c. Is the amount increasing or decreasing?
d. Suppose that instead of $\$ 10,$ a fixed amount $d$ dollars is withdrawn each month. Find a value of $d$ such that the amount in the account after each month remains $\$ 1000$ .
e. What happens if $d$ is greater than this amount?

Key Concepts

Steady-State or Equilibrium Balance

A steady-state or equilibrium balance occurs when the conditions of the system lead to a constant value over time. In practical terms, when the interest earned exactly offsets the amount withdrawn, the balance of the account remains unchanged month to month, representing a stable financial condition.

Recurrence Relations

Recurrence relations describe sequences where each term is defined as a function of its preceding term or terms. They are fundamental in modeling situations that evolve over discrete time intervals and are particularly useful for representing dynamic processes like bank account balances that change due to recurring interest calculations and withdrawals.

Compound Interest

Compound interest is the process by which an initial principal amount grows over time as interest is calculated not just on the initial sum but also on any accumulated interest from previous periods. Here, the interest is compounded on a monthly basis, which means that the effective growth factor is determined by the monthly rate applied iteratively.

Linear Difference Equations

Linear difference equations are equations that set a linear combination of previous terms equal to the next term, often including a constant term. In contexts like financial modeling, they describe processes where a quantity changes by a multiplicative factor combined with a fixed addition or subtraction, such as earning interest and making regular withdrawals.

Stability of Dynamic Systems

Stability in dynamic systems refers to whether the state of the system (like an account balance) converges to a fixed value, diverges, or oscillates over time. A stable system will tend to return to or maintain a certain equilibrium despite the ongoing changes, which is key in understanding long-term financial outcomes.

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