Series arithmetic

If the kth differences are equal, so that subsequent differences would be zero, the series is an arithmetical series of the kth order. The nth term of the series is a, and the sum of the first n terms is S, where... 

By expanding the Helmholtz free energy F at constant T in an arithmetic series in terms of ujk, we see that the linear terms vanish in view of the equilibrium condition (Euler relation for homogeneous functions of second order, F is given as... 

Several models have been proposed to estimate the thermal conductivity of hydrate/gas/water or hydrate/gas/water/sediment systems. The most common are the classical mixing law models, which assume that the effective properties of multicomponent systems can be determined as the average value of the properties of the components and their saturation (volumetric fraction) of the bulk sample composition. The parallel (arithmetic), series (harmonic), or random (geometric) mixing law models (Beck and Mesiner, 1960) that can be used to calculate the composite thermal conductivity (kg) of a sample are given in Equations 2.1 through 2.3. 

At this time, Stockman, et. al134. and Lamb135 and Baylor et. al.136 are using empirical expressions for the absorption spectrums of human chromophores. Stockman, et. al. are using a conventional arithmetic series in even powers of the variable. They make no claim to a physical foundation for their series. Lamb says It needs to be emphasized that the above (his) represents no more than an exercise in curve-fitting, and that neither equation (1) nor equation (2) has any known physical significance.. . These equations (his equation 2 in particular) are basically attempts to approximate the Helmholtz-Boltzmann equation as it is derived from the Fermi-Dirac equation, by empirical... 

In the application of the sum-of-the-years-digits method, the annual depreciation is based on the number of service-life years remaining and the sum of the arithmetic series of numbers from 1 to n, where n represents the total service life. The yearly depreciation factor is the number of useful service-life years remaining divided by the sum of the arithmetic series. This factor times the total depreciable value at the start of the service life gives the annual depreciation cost. 

Arithmetic series progress by adding (or subtracting) a constant number to each term. For example, look at the series ... 

Notice that each term is three more than the term that comes before it. Therefore, this is an arithmetic series with a common difference of 3. 

Note that the logs of the numbers in a geometric series will form an arithmetic series (e.g. 0, 1, 2, 3, 4,... in the above case). Thus, if a quantity y varies with a quantity x such that the rate of change in y is proportional to the value of y (i.e. it varies in an exponential maimer), a semi-log plot of such data will form a straight line. This form of relationship is relevant for chemical kinetics and radioactive decay (p. 236). 

As an example, consider the case of a piece of equipment eosting 20,000 when new. The service life is estimated to be 5 years and the serap value 2000. The sum of the arithmetic series of numbers from lto isl+2+3+4 +... 

Once the range of substrate concentrations to be used is chosen, the intermediate levels must be decided upon. If substrate concentrations are spaced in an arithmetic series, too many of the points will be in the high concentration range. A geometric series is better, but the best procedure is probably to space the points evenly on a reciprocal plot. 

A food dye, Orleans Blue, in aqueous solution was used to provide an arithmetic series of solutions in... 

Procedures. The principle of all of the above tests is that serial dilutions of the test disinfectant and of the standard phenol (dilutions in arithmetic series for the Rideal-Walker, F.D.A, and A.O.A.C. tests and in logarithmic series for the Chick-Martin test) are inoculated with a given amount of the culture and at intervals one 4-mm loopful of the mixture is transferred to a tube of the culture medium which is incubated at 37° and the growths noted. By equating the end-point dilution of the disinfectant with that of phenol giving the same response, the phenol coefficient of the disinfectant is obtained. 

This sum is a simple arithmetic series, which is evaluated by multiplying the number of terms by the mean of the smallest and largest terms ... 

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