Please write sequences of cars and cdrs that will pick the symbol 'x out of the following expressions: (x y z m) (y x z m) (y z m x) ((y)(x)(z)(m)) ((y z) (m x))
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Please write sequences of cars and cdrs that will pick the symbol 'x out of the following expressions:
(x y z m)
(y x z m)
(y z m x)
((y)(x)(z)(m))
((y z) (m x))
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- 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,…In C++ (Algebra: solve 2 × 2 linear equations) You can use Cramer’s rule to solve the following 2 × 2 system of linear equation: ax + by = ecx + dy = fx = (ed - bf) / (ad - bc)y = (af - ec) / (ad - bc) Write a program that prompts the user to enter a, b, c, d, e and f and displays the result. If ad - bc is 0, report that “The equation has no solution”. Sample Run 1 Enter a, b, c, d, e, f: 9.0 4.0 3.0 -5.0 -6.0 -21.0 x is -2 and y is 3 Sample Run 2 Enter a, b, c, d, e, f: 1.0 2.0 2.0 4.0 4.0 5.0 The equation has no solutionFinal Expression = x4'x3'x2'x1' + x4x3x2'x1' + x4x3'x2x1' + x4x3'x2'x1 + x4'x3x2x1' + x4'x3x2'x1 + x4'x3'x2x1 + x4x3x2x1 This expression needs simplification with boolean algebra
- Problem Description: In the Whimsical Library, each book is enchanted with a magical code to ensure the protection of knowledge. The validation process for these magical codes involves a combination of mathematical spells. The validation process is described as follows: Take the sum of the digits at odd positions from left to right. Take the sum of the squared digits at even positions from left to right. If the absolute difference between the results from step 1 and step 2 is a prime number, the book code is considered valid; otherwise, it is invalid. Example 1: Consider the magical book code 86420137. Step 1. Take the sum of the digits at odd positions from left to right. 8+4+0+3=15 Step 2. Take the sum of the squared digits at even positions from left to right. 6^2+2^2+1^2+7^2=36+4+1+49 = 90 Step 3. If the absolute difference between the results from step 1 and step 2 is a prime number. 115-901 75 (not a prime number) Book code 86420137 is invalid. Example 2: Now, let's examine the…Q1: One way to generate the value of Eulers number e is given by the formula e = lim (1 + n00 Complete the code that calculates the expression inside the limit, that is, (1 + )", for some value n. In [ 1: M # complete the function with the variable "n" and return the value as the variable named "e". # Don't define value of "n" directly inside the function, this should be an input value when you call the function # For example, approximate_e(1), the same rule applies for the rest of questions! def approximate_e(n): # your code here return e In [ ]: N ## Don't write any code in this box In [ ]: M ## Don't write any code in this box Q2: Complete the function that calculates the sum of the first n+1 terms of the geometric series a, ar, ar ar, ar*... ar" note that the series begins with a and the last term is ar", in total there are n+1 terms. In [ ]: M # complete the function given the variables a,r,n and return the value as series_sum. def sum_geometric_series(a,r,n): # your code here…(Numerical) Write a program that tests the effectiveness of the rand() library function. Start by initializing 10 counters to 0, and then generate a large number of pseudorandom integers between 0 and 9. Each time a 0 occurs, increment the variable you have designated as the zero counter; when a 1 occurs, increment the counter variable that’s keeping count of the 1s that occur; and so on. Finally, display the number of 0s, 1s, 2s, and so on that occurred and the percentage of the time they occurred.
- (Numerical) Using the srand() and rand() C++ library functions, fill an array of 1000 floating-point numbers with random numbers that have been scaled to the range 1 to 100. Then determine and display the number of random numbers having values between 1 and 50 and the number having values greater than 50. What do you expect the output counts to be?The program in the Programming Example: Fibonacci Number does not check whether the first number entered by the user is less than or equal to the second number and whether both the numbers are non-negative. Also, the program does not check whether the user entered a valid value for the position of the desired number in the Fibonacci sequence. Rewrite that program so that it checks for these things.Broken Cabins Problem Statement: There is an Office consisting of m cabins enumerated from 1 to m. Each cabin is 1 meter long. Sadly, some cabins are broken and need to be repaired. You have an infinitely long repair tape. You want to cut some pieces from the tape and use them to cover all of the broken cabins. To be precise, a piece of tape of integer length t placed at some positions will cover segments 5,5+1-sit-1. You are allowed to cover non-broken cabins, it is also possible that some pieces of tape will overlap. Time is money, so you want to cut at most k continuous pieces of tape to cover all the broken cabins. What is the minimum total length of these pieces? Input Format The first line contains three integers n,m and k(1sns10°, namsloº, Isksn) - the number of broken cabins, the length of the stick and the maximum number of pieces you can use The second line contains n integers bl,b2,bn (Isbism) - the positions of the broken cabins. These integers are given in increasing…
