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a_{n+1} = 1/16 (1 + 4a_n +\sqrt{1 + 24a_n})

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

November 21, 2022
algebrarecurrence relation

Problem Statement

The sequence a1,a2,a3,...a_1, a_2, a_3,... is defined by a1=1   ,   an+1=116(1+4an+1+24an)   (n∈N∗).a_1 = 1\,\,\,, \,\,\,a_{n+1} =\frac{1}{16}(1 + 4a_n +\sqrt{1 + 24a_n}) \,\,\,(n \in N^* ). Determine and prove a formula with which for every natural number nn the term ana_n can be computed directly without having to determine preceding terms of the sequence.
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