sec summation techniques for some constants and if this is the casewe can plug in the answers to small cases of the summation to solve for the coefficients for this examplesubstituting and for leads to three simultaneous equations because the summation when is just must be for and we get the two equations which in turn yield / and / thusif the closed-form solution for the summation is polynomialit can only be / / which is more commonly written ( at this pointwe still must do the "testpart of the guess-and-test approach we can use an induction proof to verify whether our candidate closed-form solution is correct in this case it is indeed correctas shown by example the induction proof is necessary because our initial assumption that the solution is simple polynomial could be wrong for exampleit might have been possible that the true solution includes logarithmic termsuch as log the process shown here is essentially fitting curve to fixed number of points because there is always an -degree polynomial that fits pointswe had not done enough work to be sure that we to know the true equation without the induction proof guess-and-test is useful whenever the solution is ap polynomial expression in particularsimilar reasoning can be used to solve for = or more generally pn = for any positive integer why is this not universal approach to solving summationsbecause many summations do not have polynomial as their closed form solution more general approach is based on the subtract-and-guess or divide-andguess strategies one form of subtract-and-guess is known as the shifting method the shifting method subtracts the summation from variation on the summation the variation selected for the subtraction should be one that makes most of the terms cancel out to solve sum we pick known function and find pattern in terms of (ng(nor ( )/ (