By Eli Maor

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**Example text**

Geometric mean Of n positive numbers a1, a2, . , an is the expression n √(a1a2 . . an) . 34. geometric progression A progression, or sequence, of numbers in which each number is obtained from its predecessor by multiplication by a constant number, called the quotient of the progression. Examples are 1, 2, 4, 8, 16, . , 2n – 1, . . (here the initial term is 1 and the quotient is 2), 1, 1/2, 1/4, 1/8, 1/16, . , 1/2n – 1, . . (initial term 1, quotient 1/2), and 1, –1, 1, –1, . , (–1)n – 1, . .

Bn = (1/π) –π ∫ f(x) sin nx dx, n = 1, 2, . . (note that the first formula applies also for n = 0, giving us the coefficient a0). For example, the function f(x) = x, regarded as a periodic function over [–π, π], is represented by the series 2[(sin x)/1 – (sin 2x)/2 + (sin 3x)/3 – + . ], which has only sine terms. Fourier series are used in physics to describe vibration and wave phenomena. The series is named after its discoverer, JEAN-BAPTISTE-JOSEPH FOURIER. function(s) Algebraic: See ALGEBRAIC FUNCTIONS.

The term “half-life” can be applied to any quantity that decays exponentially with time. If the quantity decays according to the formula y = y0e–kt (where y0 is the initial quantity and k a positive constant), then the half-life, denoted by the Greek letter τ (tau) is given by τ = (ln 2)/k. See also EXPONENTIAL DECAY. half-open interval An interval that is open at one endpoint and closed at the other. If the open endpoint is on the left and the closed endpoint on the right, we denote the interval by (a, b]; if the closed endpoint is on the left and the open endpoint on the right, by [a, b).