chap analysis techniques example find the closed form solution for ni= using the divideand-guess approach we will try two example functions to illustrate the divide-and-guess methoddividing by and dividing by ( our goal is to find patterns that we can use to guess closed-form expression as our candidate for testing with an induction proof to aid us in finding such patternswe can construct table showing the first few numbers of each functionand the result of dividing one by the otheras follows ( ( )/ / ( ( )/ ( / / / / / / / / / / / / / / / / / / dividing by both and ( happen to give us useful patterns to (nn+ work with (nn+ and ( - - of courselots of other approaches do not work for examplef (nn ( knowing that (nf ( is not useful for determining the closed form solution to this summation or consider (nf ( againknowing that (nf ( is not useful finding the right combination of equations can be like finding needle in haystack in our first examplewe can see directly what the closed-form solution should be (nn+ obviouslyf (nn( )/ dividing (nby ( does not give so obvious resultbut it provides another useful illustration (nn+ ( - ( )( ( ) ( ( )( ( )( (nnnf (nf (nnf (nf (nn (nn ( ( (