Growth, Decay and Series

A sum of money invested at simple interest grows by the same amount at the set intervals when interest is paid, e.g. every year or every quarter. The same amount at the same rate will grow more rapidly at compound interest because the interest due in a new period is calculated not on the original investment alone, but on the larger sum which includes interest already earnt. The first system is a particular example of the mathematical pattern known as an arithmetical progression (A.P.) made when progress from the first term (a) is made by the addition or subtraction of the same amount, known as a common difference (d): the second is a geometrical progression (G.P.) each term, as the separate values are called, being a fixed ratio (r) of the term immediately before. Progressions are a sub-class of series, sets of values generated by the substitution of different numbers in a generating formula, e.g. <math display='block'> <mrow> <mi>ln</mi><mrow><mo>(</mo> <mrow> <mn>1</mn><mo>+</mo><mi>X</mi> </mrow> <mo>)</mo></mrow><mo>=</mo><mi>X</mi><mo>&#x2212;</mo><mfrac> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> </mrow> <mn>2</mn> </mfrac> <mo>+</mo><mfrac> <mrow> <msup> <mi>X</mi> <mn>3</mn> </msup> </mrow> <mn>3</mn> </mfrac> <mo>&#x2212;</mo><mfrac> <mrow> <msup> <mi>X</mi> <mn>4</mn> </msup> </mrow> <mn>4</mn> </mfrac> <mo>&#x2026;</mo> </mrow> </math>$$\ln \left( {1 + X} \right) = X - \frac{{{X^2}}}{2} + \frac{{{X^3}}}{3} - \frac{{{X^4}}}{4} \ldots$$ is called a series, rather than a progression, because, given the numerical value of any term, we cannot conveniently calculate the next one from it, whereas in a progression we can.