If \(S_{1}, S_{2}, S_{3}\) are the sums of \(n\) terms of three arithmetic sequences, the first term of each being \(2\) and the respective common differences being \(3\), \(5\), \(7\); verify that \(S_{1} + S_{3} = 2S_{2}\).
Solution:
When \(a =\) , \(d =\)
\(S_{1} =\)
When \(a =\) , \(d =\)
\(S_{2}=\)
When \(a =\) , \(d =\)
\(S_{3}=\)
By using \(S_{1}\) and \(S_{3}\), we get \(S_{1} + S_{3}\) = \(2S_{2}\)