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- P1 P2 P3 P4 A pentagonal number is one that corresponds to dots arranged into a pentagon. Above are the first four pentagonal numbers and their pentagons. The pentagonal number PN has sides made of N dots. When you count the dots for a pentagonal number, include the dots for the inner pentagons. So to figure out how many dots PN has, find out how many P(N-1) has and add the dots for the new sides of size N. P1 = 1, P2 = 5, P3 = 12, P4 = 22 Figure out a recursive formula for PN and then write a function that implements it. That is, implement pentagonal() using recursion. pentagonal(1) -> 1 pentagonal(2) -> 5 pentagonal(3) -> 12Exercise 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.
- 4 -A mathematician and discovered a number which have unique properties. As he is not decided any name to that number, so for the time being we call that number ‘Cuboid’. A unique property of Cuboid is that, if we break that number into individual units like breaking 153 into 1, 5 and 3 and take their individual cube, then the sum would be the same as original number being given. Your task is to ask the user an integer and print weather the digit entered is Cuboid or not. (Python program)When you borrow money to buy a house, a car, or for some other purposes, then you typically repay it by making periodic payments. Suppose that the loan amount is L, r is the interest rate per year, m is the number of payments in a year, and the loan is for t years. Suppose that i = (r / m) and r is in decimal. Then the periodic payment is: You can also calculate the unpaid loan balance after making certain payments. For example, the unpaid balance after making k payments is: where R is the periodic payment. (Note that if the payments are monthly, then m = 12.) Instructions Write a program that prompts the user to input the values of L, r, m, t, and k. The program then outputs the appropriate values. Your program must contain at least two functions, with appropriate parameters, to calculate the periodic payments and the unpaid balance after certain payments. Make the program menu driven and use a loop so that the user can repeat the program for different values. Example of…A nutritionist who works for a fitness club helps members by evaluating their diets. As part of her evaluation, she asks members for the number of fat grams and carbohydrate grams that they consume in a day. Then, she calculates the number of calories that result from the fat using the following formula: Calories from fat = Fat grams × 9Next, she calculates the number of calories that result from the carbohydrates using the following formula: Calories from carbs = Carbs grams × 4Create an application that will make these calculations. In the application, you should have the following methods:• FatCalories–This method should accept a number of fat grams as an argument and return the number of calories from that amount of fat.• CarbCalories–This method should accept a number of carbohydrate grams as an argument and return the number of calories from that amount of carbohydrates.
- plz solve it asap ????????: Write a program for a game that consists of two players. The rule of the game is that player 1 inputs a random number from 1-100. Player 2 will input 5 numbers. If the summation of those numbers is equal to the number OR less than or greater than the number by 3 (+3 or -3), then player 2 wins. Otherwise, player 1 wins. Test Case 1 Input Player 1, enter a number: 50 Player 2, enter a number: 10 Player 2, enter a number: 10 Player 2, enter a number: 20 Player 2, enter a number: 5 Player 2, enter a number: 7 Output Player 2 wins Explanation: The summation of 10,10,20,5 and 7 is 52 which is greater than 50 by 2. According to the rule, player 2 wins. Test Case 2 Input Player 1, enter a number: 50 Player 2, enter a number: 10 Player 2, enter a number: 10 Player 2, enter a number: 20 Player 2, enter a number: 5 Player 2, enter a number: 1 Output Player 1 wins Explanation: The summation of 10,10,20,5 and 1 is 46 which is less than 50 and also doesn’t…In CORAL LANGUAGE please and thank you! Summary: Given integer values for red, green, and blue, subtract the gray from each value. Computers represent color by combining the sub-colors red, green, and blue (rgb). Each sub-color's value can range from 0 to 255. Thus (255, 0, 0) is bright red, (130, 0, 130) is a medium purple, (0, 0, 0) is black, (255, 255, 255) is white, and (40, 40, 40) is a dark gray. (130, 50, 130) is a faded purple, due to the (50, 50, 50) gray part. (In other words, equal amounts of red, green, blue yield gray). Given values for red, green, and blue, remove the gray part. Ex: If the input is: 130 50 130 the output is: 80 0 80 Hint: Find the smallest value, and then subtract it from all three values, thus removing the gray.Summary When you borrow money to buy a house, a car, or for some other purposes, then you typically repay it by making periodic payments. Suppose that the loan amount is L, r is the interest rate per year, m is the number of payments in a year, and the loan is for t years. Suppose that i = (r / m) and r is in decimal. Then the periodic payment is (first image which R=....): You can also calculate the unpaid loan balance after making certain payments. For example, the unpaid balance after making k payments is (second image which L'=....) : where R is the periodic payment. (Note that if the payments are monthly, then m = 12.) Instructions Write a program that prompts the user to input the values of L, r, m, t, and k. The program then outputs the appropriate values. Your program must contain at least two functions, with appropriate parameters, to calculate the periodic payments and the unpaid balance after certain payments. Make the program menu driven and use a loop so that the…
- Extreme Fizz BuzzFizz Buzz is a classic Computer Science problem about Selection. For an integer x, print:• “Fizz” if x is only divisible by 3,• “Buzz” if x is only divisible by 5,• “FizzBuzz” if x is divisible by both 3 and 5,• the number x if it is not divisible by both 3 and 5.This problem is the extreme version of Fizz Buzz. You are given a list of K distinct integers A and a list of K characters S. Both lists are numbered from 1 to K. The integer Ai corresponds to the character Si. Denote B as the subset of A. There will be a rule for each possible subset: if an integerx is only divisible by all elements in B, print a string which is the concatenation of the corresponding character of each element in B, sorted by the index in ascending order. However, if B is an empty subset, print the first digit of x instead. Does it look complicated? Do not worry, as it is just a generalization of the regular Fizz Buzz. For example, if A = [3, 5] and S = [F, B], you will get the normal Fizz…Questioin: Hidden Secret Problem! In this problem, you are given two names, you have to find whether one name is hidden into another. The restrictions are: You can change some uppercase letters to lower case and vice versa. You can add/remove spaces freely. You can permute the letters. And if both the names match exactly, then you can say that one name is hidden into another. Input Input starts with an integer T (≤ 100), denoting the number of test cases. Each case starts with two lines. Each line contains a name consists of upper/lower case English letters and spaces. You can assume that the length of any name is between 1 and 100(inclusive). Output For each case, print the case number and Yes if one name is hidden into another. Otherwise print No. Sample Input Output 3 Tom Marvolo Riddle I am Lord Voldemort I am not Harry Potter Hi Pretty Roar to man Harry and Voldemort Tom and Jerry and Harry Case 1: Yes Case 2: Yes Case 3: No And I want…7) The ideal gas law allows the calculation of volume of a gas given the pressure(P), amount ofthe gas (n), and the temperature (T). The equation is:V = nRT / PSince we only have used integer arithmetic, all numbers will be integer values with nodecimal points. The constant R is 8.314 and will be specified as (8314/1000). This givesthe same result. Implement the idea gas law program where the user is prompted for andenters values for n, T, and P, and V is calculated and printed out. Be careful toimplement an accurate version of this program. Your program should include a properand useful prompt for input, and print the results in a meaningful manner