100 jumps

An infinite line of stepping stones stretches out into an infinitely large lake.

A frog starts on the second stone.

Every second he takes a jump to a neighboring stone. He has a probability, $$p$$, of jumping one stone further away from the shore and a probability, $$1-p$$, of jumping one stone closer to the shore.

If the expected value for the number of jumps he will take before reaching the first stone is exactly 100, then $$p$$ can be expressed as $$\dfrac ab$$, where $$a$$ and $$b$$ are coprime positive integers, find $$a+b$$.

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