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Two Sequences

Source: 1996 National High School Mathematics League, Exam Two, Problem 1

March 4, 2020

Problem Statement

The sum of first nn items of squence (an)(a_n) : SnS_n satisfies that Sn=2an−1S_n=2a_n-1, squence (bn)(b_n) satisfies that bk+1=ak+bkb_{k+1}=a_k+b_k for all k=1,2,⋯k=1,2,\cdots. Find the sum of first nn items of (bn)(b_n).
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