app6 - Poularikas, A.D. Appendix 6: Algebra Formulas and...

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Poularikas, A.D . Appendix 6 : A lgebra Formulas and Coordinate Systems." The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000
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© 2000 by CRC Press LLC Appendix 6: Algebra Formulas and Coordinate Systems Arithmetic Progression * An arithmetic progression is a sequence of numbers such that each number differs from the previous number by a constant amount, called the common difference. If a 1 is the Frst term, a n the n th term, d the common difference, n the number of terms, and s n the sum of n terms. The arithmetic mean between a and b is given by . Geometric Progression * A geometric progression is a sequence of numbers such that each number bears a constant ratio, called the common ratio , to the previous number. If a 1 is the Frst term, a n the n th term, r the common ratio, n the number of terms, and s n the sum of n terms. * It is customary to represent a n by l in a Fnite progression and refer to it as the last term. aands n aa s n and nn n n =+− () =+ [] 11 1 1 22 21 ,, . ab + 2 r sa r r a r r r ar a r ra a r n n n n n n n == = = = 1 1 1 1 1 1 1 1 1 1 1 1 1 ;
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© 2000 by CRC Press LLC If * r * < 1, then the sum of an inFnite geometrical progression converges to the limiting value The geometric mean between a and b is given by . Harmonic Progression A sequence of numbers whose reciprocals form an arithmetic progression is called an harmonic progres- sion. Thus, where forms a harmonic progression. The harmonic mean between a and b is given by . If A , G H , respectively, represent the arithmetic mean, geometric mean, and harmonic mean between a and b , then G 2 = AH . Factorials Permutations If M = n P r = P n:r denotes the number of permutations of n distinct things taken r at a time, Combinations If M = n C r = C = denotes the number of combinations of n distinct things taken r at a time, a r s ar r a r n n 1 1 1 1 1 11 = () = →∞ , lim ab 1 2 1 1 1 1 aada d and , , ,, ++ +− KK 1 1 a n = 2 ab ab + ∠= = nn en n nn !, . 2 π approximately Mn n n nr n =− −+ = 12 1 L ! ! r n M n n r r n rn r = = 1 L ! ! !!
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© 2000 by CRC Press LLC By defnition = 1. Quadratic Equations Any quadratic equation may be reduced to the Form, ax 2 + bx c = 0 . Then IF a , b , and c are real, then: IF b 2 – 4 ac is positive, the roots are real and unequal. IF b 2 – 4 ac is zero, the roots are real and equal. IF b 2 – 4 ac is negative, the roots are imaginary and unequal. Cubic Equations A cubic equation, y 3 + py 2 + qy + r = 0 may be reduced to the Form, x 3 b by substituting For y the value, . Here ±or solution, let then the values oF x will be given by, IF p q r are real, then: IF > 0, there will be one real root and two conjugate imaginary roots. IF = 0, there will be three real roots oF which at least two are equal.
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app6 - Poularikas, A.D. Appendix 6: Algebra Formulas and...

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