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I want to get the exact form of general formulas, but RSolve only gives a DifferenceRoot object. I tried FunctionExpand but it does not work.

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You can get the generic solution easily with the substitution a[n]->101 Log[b[n]]. The difficult bit seems to be matching your initial condition, though a numerical solution is found quickly. –  b.gatessucks May 19 at 8:52

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