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Find explicit function for recursively defined function

Source: Germany Bundeswettbewerb Mathematik 2001, Round 1, Problem 2

February 1, 2009
functionquadraticsalgebra unsolvedalgebra

Problem Statement

For a sequence ai∈R,i∈{0,1,2,…} a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\} we have a_0 \equal{} 1 and a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n}   \forall n \in \mathbb{N}. Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.
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