Task 1. Find the recursive formula for the sequence 2, 3, 6, 18, 108, 1944, ......

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Task 1. Find the recursive formula for the sequence 2, 3, 6, 18, 108, 1944, ... Then define a recursive function named mysterySequence that returns the nth member of the sequence. For example, the output for mysterySequence (4) is 108. Task 2. Spherical objects, such as tennis balls, can be stacked to form a pyramid with one tennis ball at the top, sitting on top of a square composed of four tennis balls, sitting on top of a square composed of nine tennis balls, and so forth. Write a recursive function that takes as its argument the height of a pyramid of tennis balls and returns the number of tennis balls it contains. For example: tennisBalls (3) returns 14. Task 3. Define a recursive function named decToBase2 which given an integer, returns its binary representation as a string. For example, the binary equivalent of 13 may be found by repeatedly dividing 13 by 2. So, 13 in base 2 is represented by the string "1101" and the output for decToBase2 (13) should be "1101". Task 4. Define a recursive function isDivisibleBy7 that returns true if the given number is divisible by 7, and false otherwise. For example: isDivisibleBy7(1073) returns false. To find out if a number is divisible by 7, you can follow this algorithm: Remove the last digit and double it. Subtract it from the remaining number. If the result is zero or a recognizable 2-digit multiple of 7, then the number is divisible by 7. Otherwise no or repeat if more than 2 digits are left. For example: Is 1073 divisible by 7? Remove the last digit, 3, from the number and double it, which becomes 6. The remaining number becomes 107, so 107-6 = 101. 101 has 3 digits (which is more than 2), so repeating the process one more time: remove the last digit, 1, and double it, which becomes 2. Remaining number 10-2-8. As 8 is not divisible by 7, the number 1073 is not divisible by 7. Your function *must be recursive and implement the above algorithm. You cannot use division or modulo to check for divisibility by 7 until you are down to a 2-digit number. Task 5. The famous Indian mathematician, Srinivasa Ramanujan, asked a question that stumbled a number of people: what is the value of 6+2 7+3 8+4 9+510+- carried out to infinity? Define a function, named ramanujan. Which takes, as one of its arguments, the depth of a rational approximation to the above nested expression. For example, if the depth is 0, Ramanujan should return 6. If the depth is 1, Ramanujan should return the value of 6 +27. If the depth is 2, the return value should be the value of 6 +27 + 33. Your function should implement a recursive process. Your function may have more than one argument, as needed for your recursion to work. At the end of your code, you need to give the value of the above expression when carried out to infinite. Task 6. Write a function that tests all your other functions (Task 1-5) and neatly prints out their results. The following tests should be run for each and their results printed to screen. 1. mysterySequence with n = 0, 1, 4, 7 2. tennisBalls with height = 0, 1, 3, 10 3. decToBase2 with input = 0, 1, 13, 32, 321 4. isDivisibleBy7 with input = 1, 7, 31, 1073, 1729 5. Ramanujan with depth = 1, 3, 10 These tests are not exhaustive, and I suggest you run many more tests on your own to check the functionality of each of your functions from Task 1-5. Welcome to the recursion assignment. What would you like to test? 1. mysterySequence 2. tennisBalls 3. decToBase2 4. isDivisibleBy7 5. ramanujan 6. run tests 7. Exit 1 Mystery Sequence. Please enter n to compute the nth number in sequence: 4 The 4th number in mystery sequence is 108. Welcome to the recursion assignment. What would you like to test? 1. mysterySequence 2. tennisBalls 3. decToBase2 4. isDivisibleBy7 5. ramanujan 6. run tests 7. Exit 2 Tennis Balls. Please enter the height of the pyramid: 3 A pyramid with 3 levels holds 14 tennis balls. welcome to the recursion assignment. What would you like to test? 1. mysterySequence 2. tennisBalls 3. decToBase2 4. isDivisibleBy7 5. ramanujan run tests 7. Exit Ramanujan. Enter integer depth: 7 Result at depth 7: ????? Result at infinite depth: ????? Task 1. Find the recursive formula for the sequence 2, 3, 6, 18, 108, 1944, ... Then define a recursive function named mysterySequence that returns the nth member of the sequence. For example, the output for mysterySequence (4) is 108. Task 2. Spherical objects, such as tennis balls, can be stacked to form a pyramid with one tennis ball at the top, sitting on top of a square composed of four tennis balls, sitting on top of a square composed of nine tennis balls, and so forth. Write a recursive function that takes as its argument the height of a pyramid of tennis balls and returns the number of tennis balls it contains. For example: tennisBalls (3) returns 14. Task 3. Define a recursive function named decToBase2 which given an integer, returns its binary representation as a string. For example, the binary equivalent of 13 may be found by repeatedly dividing 13 by 2. So, 13 in base 2 is represented by the string "1101" and the output for decToBase2 (13) should be "1101". Task 4. Define a recursive function isDivisibleBy7 that returns true if the given number is divisible by 7, and false otherwise. For example: isDivisibleBy7(1073) returns false. To find out if a number is divisible by 7, you can follow this algorithm: Remove the last digit and double it. Subtract it from the remaining number. If the result is zero or a recognizable 2-digit multiple of 7, then the number is divisible by 7. Otherwise no or repeat if more than 2 digits are left. For example: Is 1073 divisible by 7? Remove the last digit, 3, from the number and double it, which becomes 6. The remaining number becomes 107, so 107-6 = 101. 101 has 3 digits (which is more than 2), so repeating the process one more time: remove the last digit, 1, and double it, which becomes 2. Remaining number 10-2-8. As 8 is not divisible by 7, the number 1073 is not divisible by 7. Your function *must be recursive and implement the above algorithm. You cannot use division or modulo to check for divisibility by 7 until you are down to a 2-digit number. Task 5. The famous Indian mathematician, Srinivasa Ramanujan, asked a question that stumbled a number of people: what is the value of 6+2 7+3 8+4 9+510+- carried out to infinity? Define a function, named ramanujan. Which takes, as one of its arguments, the depth of a rational approximation to the above nested expression. For example, if the depth is 0, Ramanujan should return 6. If the depth is 1, Ramanujan should return the value of 6 +27. If the depth is 2, the return value should be the value of 6 +27 + 33. Your function should implement a recursive process. Your function may have more than one argument, as needed for your recursion to work. At the end of your code, you need to give the value of the above expression when carried out to infinite. Task 6. Write a function that tests all your other functions (Task 1-5) and neatly prints out their results. The following tests should be run for each and their results printed to screen. 1. mysterySequence with n = 0, 1, 4, 7 2. tennisBalls with height = 0, 1, 3, 10 3. decToBase2 with input = 0, 1, 13, 32, 321 4. isDivisibleBy7 with input = 1, 7, 31, 1073, 1729 5. Ramanujan with depth = 1, 3, 10 These tests are not exhaustive, and I suggest you run many more tests on your own to check the functionality of each of your functions from Task 1-5. Welcome to the recursion assignment. What would you like to test? 1. mysterySequence 2. tennisBalls 3. decToBase2 4. isDivisibleBy7 5. ramanujan 6. run tests 7. Exit 1 Mystery Sequence. Please enter n to compute the nth number in sequence: 4 The 4th number in mystery sequence is 108. Welcome to the recursion assignment. What would you like to test? 1. mysterySequence 2. tennisBalls 3. decToBase2 4. isDivisibleBy7 5. ramanujan 6. run tests 7. Exit 2 Tennis Balls. Please enter the height of the pyramid: 3 A pyramid with 3 levels holds 14 tennis balls. welcome to the recursion assignment. What would you like to test? 1. mysterySequence 2. tennisBalls 3. decToBase2 4. isDivisibleBy7 5. ramanujan run tests 7. Exit Ramanujan. Enter integer depth: 7 Result at depth 7: ????? Result at infinite depth: ?????