Геометрическая прогрессия a1+a2+...+an = s 1/a1+1/a2+....1/an= t найти a1*a2*a3*...*an

 [latex] a_{1}+a_{2}+...+a_{n} =S\\ \frac{1}{a_{1}} + \frac{1}{a_{2}} + ... + \frac{1}{a_{n}} = t \\\\ [/latex]     Положим что   [latex]q\ \textgreater \ 1\\ a_{1}+a_{1}q+a_{1}q^2+...+a_{1}q^{n-1} = \frac{a_{1}(q^n-1)}{q-1}=S \\ \frac{1}{a_{1}} + \frac{1}{a_{2}}+...+\frac{1}{a_{n} } = \frac{1}{a_{1}}(1+\frac{1}{q}+...) =\\ \frac{1}{a_{1}} * \frac{ (\frac{1}{q})^n-1}{\frac{1}{q}-1} = \frac{ q - q^{1-n}} { a_{1}(q-1) } = t \\ \\ \frac{a_{1}(q^n-1)}{q-1} = S\\ [/latex]       [latex] \frac{q-q^{1-n}}{a_{1}(q-1)} = t \\\\ a_{1}a_{2}a_{3}*...*a_{n} = a_{1}^{n}*q^{ \frac{n^2-n}{2}}\\ a_{1}^2*q^{n-1} = \frac{S}{t}\\ a_{1}^n*q^{\frac{(n-1n}{2}} = (\frac{S}{t})^{\frac{n}{2}} [/latex]