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- Evaluate the following expressions if x is 10.5, y is 7.2, m is 5, and n is 2. x / (double)m = x / m = (double)(n * m) = (double)(n / m) + y = (double)(n / m) =Write an expression that obtains a random integer between 34 and 55. Write an expression that obtains a random integer between 0 and 999.Exercise III: Catalan numbers For n e N, denote by c, to be the number of ways to form a "mountain range" with n upstrokes (U) and n downstrokes (D) that all stay above a horizontal line. For instance: for n = 1, only UD is allowed, so c = 1; • for n = 2, only UUDD and UDUD are allowed, so c2 = 2. 1. Check that c3 = 5 by writing down or drawing all possible options. 2. Consider the power series +00 g(x) = 2 n=0 (by definition co = 1) called the generating function of the sequence (en). Justify that cn < 4" for each n, and deduce that the radius of convergence of g is at least 1/4. 3. It can be show that for r E (-1/4, 1/4), g(x) = 1+ xg(r)? and therefore 1- VI 4.x g(x) = 2.x Use this formula and the known power series of V1+ x to write the first terms of the power series expansion of g, in the form g(x) = co + c1x + c2x2 + C3x + c4x* + ·.. Show how you obtain a few terms, but you do not need to show all computations and you can use a calculator for fractions. Check that you recover co,…
- e11: Find the value of A for the following: A = (5 + 2* 3+ (( 3-2)* 7)+ -9) /2.Q2/ Simplify the expressions using Boolean postulates: F=AB+(AC) '+ AB'C (AB+ C).Write an expression that obtains a random integer between 34 and 55. Write an expression that obtains a random integer between 0 and 999. Write an expression that obtains a random number between 5.5 and 55.5.