Write a recursive function in f#, named specialSum, that has the followeing signature: int * int -> int, where sum(m,n) = m + (m +1) + (m+2) + ... + (m + (n-1)) + (m+n) for m >= 0 and n >= 0: (Hint use two clauses with (m,0) and (m,n) A PATTERNS.) start code with let rec specialSum (m,n) match m,n with | m,0 -> | m,n - >

Define a recursive function named trim; it takes as its arguments a linked list (the head node of it) and a non-negative integer representing the index of an element in the list. The function mutates the linked list such that all elements that occur after the index is removed. Additionally, the function returns a linked list of all the elements that were moved. If the index value exceeds the length of the linked list, it should raise an IndexError. For example, if we defined a list as l1 = LN(1, LN(2, LN 3))) , calling the function would return the linked list LN(2, LN(3)), and the mutated l1 would become LN(1). You must use recursion. def trim(l: LN, index: int) -> LN:

Write a C++ program using classes and recursion functions to covert a number in a given base (the user enters the base - from 2 to 9) to a decimal number. The algorithm states that each successive digit in the number is multiplied by the base raised to the power corresponding to its position in the number. The low-order digit is in position 0. We sum together all of these products to get the decimal value. For example, in binary- base 2, if we have the binary number 111001, we convert it to decimal as follows: 1x25 + 1x2* + 1x2³ + 0x2²+ 0x2² + 1x2º = 57 For a base 5 number, 14230: 1x5 + 4x53 + 2x52 + 3x5+ Ox5° = 1190 Validate your input. You should check for invalid digits and provide an informative error message to the user. This program requires you to use a recursive function for the conversion to decimal. Failure to do that will result in no points awarded.

Give a recursive definition for the set of all strings of a’s and b’s that begins with an a and ends in a b. Say, S = { ab, aab, abb, aaab, aabb, abbb, abab..} Let S be the set of all strings of a’s and b’s that begins with a and ends in a b. The recursive definition is as follows – Base:... Recursion: If u ∈ S, then... Restriction: There are no elements of S other than those obtained from the base and recursion of S.