Geometric sequence

Figure 6.4 shows a third and commonly used way of representing electron density profiles the two-dimensional contour map. This map for the SCI2 molecule corresponds to the relief map in Figure 6.2. Although this map is able to show very detailed information, we are restricted to a particular choice of plane, or to a selection of planes. To obtain an approximately equally dense distribution of contour lines, contour values used in this book increase in the nearly geometrical sequence, I0 3, 2 X 10 3, 4 X 10 3, 8 X 10-3. 2 X 10-3.

Notice that each successive term is found by multiplying the prior term by 2. (2 X 2 = 4,4 X 2 = 8, and so on.) Since each term is multiplied by a constant number (2), there is a constant ratio between the terms. Sequences that have a constant ratio between terms are called geometric sequences. 

On the SAT, you may be asked to determine a specific term in a sequence. For example, you may be asked to find the thirtieth term of a geometric sequence like the previous one. You could answer such a question by writing out 30 terms of a sequence, but this is an inefficient method. It takes too much time. Instead, there is a formula to use. Let s determine the formula ... 

The generic formula for a geometric sequence is Term n = o1Xr" 1, where n is the term you are looking for, flj is the first term in the series, and r is the ratio that the sequence increases by. In the above example, n = 30 (the thirtieth term), ax = 2 (because 2 is the first term in the sequence), and r = 2 (because the sequence increases by a ratio of 2 each term is two times the previous term). 

You can use the formula Term fi = u1Xr "1 when determining a term in any geometric sequence. 

As shown in Fig. 1.8, the bifurcation points occur more and more frequently as A —> oo. It was Feigenbaum s great discovery (Feigenbaum (1978, 1979)) that the sequence of bifurcation points approaches a geometric sequence for A —> oo such that... 

Equations (8.21) are a set of linear equations which express the desired values of F, explicitly in terms of the measured values of Q,. The coefficients of the equations depend upon the values of (corresponding to the value of t ) at which the concentrations are measured more exactly, the coefficients depend on the ratios of the values of as shown in equation (8.22). Consequently, if the values of d, are chosen in a geometric sequence when carrying out a particle size analyses, the coefllcients are considerably easier to calculate and the equations... 

The expression between brackets is a sum of geometric sequences and therefore... 

Complex surfaces, are characterized by their strong misorientation with respect to compact planes. As a consequence the atomic steps are very closely spaced and the TLK model is not readily applied. The absenee of well-defined geometric sequences prevents a simple description of complex surfaces. 

Fibonacci Series Geometric sequence described by Italian mathematician Leonardo of Pisa (known as Fibonacci) in about 1200, beginning with zero and in which each subsequent number is the sum of the previous two 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... 

Decaying geometric sequence with alternating signs... 

---