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An easy problem about a sequence

Source: 2023 China South-east Mathematics Olympiad Grade 10 P1

July 31, 2023

algebraSequence

Problem Statement

The positive sequence

\{a_n\}

satisfies:

a_1=1

and

a_n=2+\sqrt{a_{n-1}}-2 \sqrt{1+\sqrt{a_{n-1}}}(n\geq 2)

Let

S_n=\sum\limits_{k=1}^{n}{2^ka_k}

. Find the value of

S_{2023}