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An approximation algorithm is an algorithm that finds an approximate solution to a (typically NP-hard) problem. The quality of the algorithm is measured by how close to the actual optimum it performs. For example, it is a constant factor approximation algorithm if it always outputs a solution that is within a constant factor of the optimum. Hardness of approximation is one way to separate NP-hard problems.

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Giving a general term of a recursive function, and upper bound for it

There's no general upper bound. Suppose $p_t<1$ for every $t$ and $\sum_{t=1}^\infty p_t = \infty$. For every $N$ there's a positive probability that $l_N = 0$, then $l_t$ will be larger than $NB …
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