CMR với ∀ n ∈ N* ta luôn có
\(S=\dfrac{1}{1+a_1+a_1a_2+a_1a_2a_3+...+a_1a_2a_3...a_{n-1}}\)
\(+\dfrac{1}{1+a_2+a_2a_3+a_2a_3a_4+...+a_2a_3a_4...a_n}\)
\(+\dfrac{1}{1+a_3+a_3a_4+a_3a_4a_5+...+a_3a_4a_5...a_na_1}\)
\(+...+\dfrac{1}{1+a_n+a_na_1+a_na_1a_2+a_na_1a_2...a_{n-2}}=1\)với \(a_1a_2a_3a_4...a_n=1\)
\(S=\dfrac{1}{1+a_1+a_1a_2+...+a_1a_2...a_{n-1}}+\dfrac{1}{1+a_2+a_2a_3+...+a_2a_3...a_n}+...+\dfrac{1}{1+a_n+a_na_1+...+a_na_1...a_{n-2}}\)
\(=\dfrac{1}{1+a_1+a_1a_2+...+a_1a_2...a_{n-1}}+\dfrac{1}{1+a_2+a_2a_3+...+a_2a_3...a_{n-1}+\dfrac{1}{a_1}}+...+\dfrac{1}{1+a_{n-1}+\dfrac{1}{a_1a_2...a_{n-2}}+...+\dfrac{1}{a_{n-2}}}+\dfrac{1}{1+\dfrac{1}{a_1a_2...a_{n-1}}+\dfrac{1}{a_2a_3...a_{n-1}}+...+\dfrac{1}{a_{n-1}}}\)\(=\dfrac{1}{1+a_1+a_1a_2+...+a_1a_2...a_{n-1}}+\dfrac{a_1}{1+a_1+a_1a_2+...+a_1a_2...a_n}+...+\dfrac{a_1a_2...a_{n-2}}{1+a_1+a_1a_2+...+a_1a_2...a_{n-1}}+\dfrac{a_1a_2...a_{n-1}}{1+a_1+a_1a_2+...+a_1a_2...a_{n-1}}\)
\(=\dfrac{1+a_1+a_1a_2+...+a_1a_2...a_{n-1}}{1+a_1+a_1a_2+...+a_1a_2...a_{n-1}}=1\)
