Double sequences with GCD

Given a sequence called \(A\) including \(50\) terms ( from \(1\) to \(50\)), which can be described as \[{ A }_{ n }=20+{ n }^{ 2 }\]

Given another sequence called \(B\) including \(49\) terms, which have a property that \[{ B }_{ n }=GCD({ A }_{ n },{ A }_{ n+1 })\]

Given that the largest term in \(B\) is \({B}_{x}\). Find the value of \(\sqrt { { B }_{ x }+x } \).

Note: \(GCD(a,b)\) is denoted as the greatest common divisor of \(a\) and \(b\).

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