Handbook of Applied Mathematics: Selected Results and by Prof. H. Lennart Pearson (auth.), Carl E. Pearson (eds.)

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By Prof. H. Lennart Pearson (auth.), Carl E. Pearson (eds.)

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X n - I . , then T2 , T 3 , ••• , Tn-I are called geometric means between TI and Tn. 9 BINOMIAL THEOREM. n-3 3! u 3 b + ... 9-1) with the general term, the (r + I)st term (r = 0, 1,2, ... 9-2) where n! = n . (n - 1)· (n - 2)· (n - 3) ... 3·2· 1, O! 2 The Binomial Series The infinite series 1 + gx + g(g - 1) 2! x2 + g(g - 1)(g - 2) 3! converges to (1 + x)g for all x such that Ix x3 + g(g - 1)(g - 2)(g - 3) 4! X4 I < 1. 3 Permutations. Definition and Formulas The fundamental principle of permutations and combinations is: if one act can be done in nl different ways, and upon its completion a second act can be done in n2 Formulas from Algebra, Trigonometry and Analytic Geometry 23 different ways, and upon its completion a third act can be done in n3 different ways, and so on finitely, then the total number of ways in which all the acts can be done in the order stated is n 1 • n2 .

2) In a homogeneous system, if the rank of the coefficient matrix is equal to the number of unknowns, then the solution is unique (and trivial) and is Xi = 0 for all i. If this rank is less than the number of unknowns, then an infinity of solutions exists. For example, in the set of equations solved before by elimination, the rank Formulas from Algebra, Trigonometry and Analytic Geometry 21 of the augmented matrix equals the rank of the coefficient matrix equals 3, which is less than the number of unknowns; there were indeed an infinite number of solutions.

2, 37°) is the same point as (2, 397°), or (2, - 323°). 114) Y= 1 +r 42 Handbook of Applied Mathematics y P (x. x Fig. 11·2 Polar coordina tes. 1 Angle of Inclination, Slope, Intercepts The angle of inclination of a line L is the smallest angle obtained when the + x·axis is rotated counterclockwise into coincidence with the line. Thus 0 ,;;;; a < 1800 • The slope of L is defined to be tan a and will be denoted by s. If PI (XI, YI), P2 (X2, Y2) are two points on L, then Y2 - YI X2 - XI s=--- r ---------'~"'-=+_----------II.

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