chap mathematical preliminaries because we have merely guessed at pattern and not actually proved that this is the correct closed form solutionwe can use an induction proof to complete the process (see example example slightly more complicated recurrence is (nt( nt ( expanding this recurrence few stepswe get (nt( ( ( ( ( ( we should then observe that this recurrence appears to have pattern that leads to (nt( ( )( ( )( ( this is equivalent to the summation ni= ifor which we already know the closed-form solution techniques to find closed-form solutions for recurrence relations are discussed in section prior to recurrence relations are used infrequently in this bookand the corresponding closed-form solution and an explanation for how it was derived will be supplied at the time of use recursion an algorithm is recursive if it calls itself to do part of its work for this approach to be successfulthe "call to itselfmust be on smaller problem then the one originally attempted in generala recursive algorithm must have two partsthe base casewhich handles simple input that can be solved without resorting to recursive calland the recursive part which contains one or more recursive calls to the algorithm where the parameters are in some sense "closerto the base case than those of the original call here is recursive java function to compute the