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As proof by induction goes we always have to show that it works for the base case, which in this case is the very first positive integer:1. So we show that the sum of the first number(s) is equal to one w...

3sinh^2(2x) + 11sinh(2x) - 4 = 0 --> (3sinh(2x) - 1)(sinh(2x) + 4) = 0 --> sinh(2x) = 1/3, sinh(2x) = -4(e^(2x) - e^(-2x))/2 = 1/3 --> e^(4x) -(2/3)e^(2x) - 1 = 0 --> e^(2x) = 1/3 + 2sqrt(5)/3...

This identity can easily be shown by asking the student to draw a right-angled triangle, with one of the remaing two angles labelled x, and labelling the respective oppposite, adjacent and hypotenuse side...

(1+2i) (2+i)/(2+i)(2-i) = (2-2+4i+i)/5 = 5i/5 = i

8,5,6Add '-x-' to each of the three terms on the leading diagonalCalculate the determinant of the 3x3 matrix and set equal to zero (property of eigenvalues)Factorise to find values of x, for which the det...