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Replace in recursive equations

The last sentence of my previous message about Inverse got somehow
scrambled. I meant "I have to re-input the output to get the result." 

One more quick question. Suppose I have a recursive equation F[n+1]==a
F[n]. I want to replace F[n]->FF[n]+Q for _all_ n's. Q is a constant.
What is the best way to do it? The left side should become FF[n+1]+Q
and the right a(FF[n]+Q). Any help would be appreciated. 

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