Just like Fibonacci

Algebra Level 5

Let \(\{a_r\}\) be a sequence of positive integers such that \(a_r = a_{r-1} + a_{r-2}\) for \(r\ge3\) and \(a_2=4\). It is given that \(a_n+2a_{n-1}+a_{n-2}=123\) (such that \(a_{n+2} \)is the last term of the sequence). Find the value of \[\frac{1}{na_1}\sum _{ r=1 }^{ r=n }{ a_r } +\frac{4}{a_1}\]

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