回復 1# kggj5220 的帖子
\(\begin{align}
& {{a}_{1}}=1 \\
& {{a}_{n}}={{S}_{n}}-{{S}_{n-1}}=2{{a}_{n}}+{{\left( -1 \right)}^{n}}-2{{a}_{n-1}}-{{\left( -1 \right)}^{n-1}} \\
& {{a}_{n}}=2{{a}_{n-1}}-2\times {{\left( -1 \right)}^{n}} \\
& {{a}_{n}}+\frac{2}{3}\times {{\left( -1 \right)}^{n}}=2{{a}_{n-1}}-\frac{4}{3}\times {{\left( -1 \right)}^{n}} \\
& {{a}_{n}}+\frac{2}{3}\times {{\left( -1 \right)}^{n}}=2\left[ {{a}_{n-1}}+\frac{2}{3}\times {{\left( -1 \right)}^{n-1}} \right] \\
\end{align}\)
\({{a}_{n}}+\frac{2}{3}\times {{\left( -1 \right)}^{n}}\)是首項\(\frac{1}{3}\)，公比2的等比數列
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