How can algebraic expressions represent interest earned on an investment?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

Algebraic expressions are used to model the relationship between an initial investment, the payments received, and the rate of interest. By representing these financial variables as algebraic terms, one can determine the growth of capital over time.1

For a transaction involving a present value ( $P_{0}$ ) and a series of payments ( $a_{1}, a_{2}, \dots, a_{n}$ ) received over $n$ years, the interest rate ( $r$ ) is expressed through the following equation:

$P_{0} = \frac{a_{1}}{(1 + r)^{1}} + \frac{a_{2}}{(1 + r)^{2}} + \dots + \frac{a_{n}}{(1 + r)^{n}}$

In specific scenarios, these expressions simplify:

Promissory Note: For a single payment ( $a_{n}$ ) received in the future, the expression becomes $P_{0} = \frac{a_{n}}{(1 + r)^{n}}$ .1
Perpetuity: For an investment that pays a constant annual sum ( $a$ ) indefinitely, the formula simplifies to $P_{0} = \frac{a}{r}$ .1

These algebraic models demonstrate that the rate of interest functions as a rate of growth for capital within a contractual obligation, where the market price of the bond inversely affects the resulting interest rate.1