Note! It is very important to the pedagogical goals of this studio that you follow the directions below carefully. The object of studio is not to arrive at a solution, but rather to understand the processes involved in developing solutions.

For example, at times, you may be asked to sketch a solution using pencil and paper.

To be blunt, this means you must have pencil and paper out, and you must use said paper and pencil to sketch solutions. These sketches are very helpful both for your own understanding of recursive computations but also these sketches help us understand how you think about recursion.

Overview

• A recursive specification or formula may be provided for a computation. In such cases, the recursive code is often a direct transcription of the formula into Java.
• Examples or instances of the computation may be provided. In such cases, you must identify the recursive structure of the computation to obtain code.

Recursive Formulae

1. For each of the following recursive formulae, write the corresponding code in Java and test your solution using the provided JUnit test.

   The functions defined below are partial, in the sense that they do not work on all inputs. Let's assume that inputs to all functions below are restructed to the nonnegative integers (0, 1, 2, …).

   Do not worry about errors that may occur if improper inputs are presented.

   factorial

   This was done in front of you during lecture.

   fact(n) =	{	n × fact(n-1)	n > 0
   		1	otherwise

   Fibonacci

   fib(n) =	{	fib(n-1) + fib(n-2)	n > 1
   		n	otherwise

   isOdd

   isOdd(n) =	{	! isOdd(n-1)	n > 0
   		false	otherwise

   sum

   sum(a,b) =	{	sum(a+1, b-1)	b > 0
   		a	otherwise

2. Using the substitution model, show (using pencil an paper) the evaluation of:
   • isOdd(3)
   • sum(100,3)

   At this piont, show your work to a TA before proceeding!

Recursive Substructure

After identifying the substructure, implement the function recursively in Java and run the unit test.

Show your recursive structure diagrams to a TA for each problem below before coding it or before proceeding to the next problem.

factorial

This was done in front of you during lecture.

• fact(n) = n × (n-1) × (n-2) × … × 1
• fact(0) = 1

sumDownBy2

• sumDownBy2(n) = n + (n-2) + (n-4) + …
• sumDownBy2(1) = 1
• sumDownBy2(0) = 0

harmonicSum

• harmonicSum(n) = 1 + 1/2 + 1/3 + … + 1/(n-1) + 1/n
• harmonicSum(1) = 1

NB: The result of this computation should be a double. Be careful about integer divsion!

mult

• mult(a,b) =		a + a +	…	a + a
  	|	Added together b times			|
  mult(a,0) =	0

Submitting your work (read carefully)

• If your studio contains a feedback.txt file, respond to the questions and supply any requested information.
• You must commit all of your work to your repository. It's best to do this from the top-most level of your repository, which bears your name and student ID.
• Follow the instructions in the green box below to receive credit for your work.

When you done with this studio, you must be cleared by the TA to receive credit.

• Commit all your work to your repository
• Fill in the form below with the relevant information
• Have a TA check your work
• The TA should check your work and then fill in his or her name
• Click OK while the TA watches

Studio repo name: studio6- (from your sticker)

People on your team:

	Last name	6-digit ID
1
2
3
4
5
6

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TA: Password: