A bank offers 5 % interest compounded yearly. Suppose you...

A bank offers $5 \%$ interest compounded yearly. Suppose you deposit $\$ 120$ in the account each year. Set up an affine iterative equation $(9.5)$ to represent your bank balance. How much money do you have after 10 years? After you retire in 50 years? After 200 years?

Key Concepts

Compound Interest

Compound interest refers to the process by which interest earned on an initial principal amount is reinvested to earn additional interest in subsequent periods. Instead of earning interest solely on the original amount, each period’s interest is added to the principal, and the new sum earns interest in the following period. This compounding effect leads to exponential growth of the investment over time.

Affine Iterative Equation

An affine iterative equation is a type of recurrence relation where the next term is found by multiplying the current term by a constant and then adding another constant. In financial contexts, it describes a situation where an account balance increases both by earning compound interest (the multiplicative factor) and by making regular deposits (the additive constant). This results in a pattern of growth that can be optimally described using an affine transformation.

Recurrence Relation

A recurrence relation is an equation that recursively defines a sequence: each term is formulated as a function of preceding terms. In many financial problems, like those involving compound interest combined with regular deposits, recurrence relations help model the evolving balance over time by establishing a rule that connects one time period to the next.

Annuity and Geometric Series

An annuity involves a series of equal payments made at regular intervals, and when these deposits are combined with compound interest, the total value of the annuity can be calculated using the geometric series formula. This understanding helps convert the iterative process into a closed-form solution, summarizing the cumulative effect of both interest accumulation and regular deposits over a period.

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