Exponential growth

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value, resulting in its growth with time being an exponential function. Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time.

The formula for exponential growth of a variable x at the growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is

where x₀ is the value of x at time 0. This formula is transparent when the exponents are converted to multiplication. For instance, with a starting value of 50 and a growth rate of r = 5% = 0.05 per interval, the passage of one interval would give ${\displaystyle 50(1.05)^{1}}$, or simply 50×1.05; two intervals would give ${\displaystyle 50(1.05)^{2}}$, or simply 50×1.05×1.05; and three intervals would give ${\displaystyle 50(1.05)^{3}}$, or simply 50×1.05×1.05×1.05. In this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent. (The order of multiplication does not change the result based on the associative property of multiplication.)

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