Reciprocal fraction

Algebra Level 3

Consider the sequence a1=3a_{1}=3, a2a_{2}, a3,a_{3} , \ldots, where 1ak+1=1a1+1a2++1ak for all integers k1.\dfrac{1}{a_{k+1}}=\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{k}} \text{ for all integers } k \ge 1.

Find a2016a_{2016}.

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