\[\boxed{\text{634.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[x_{1} = 2,\ \ x_{5} = x_{1} \cdot q^{4} = 162,\ \ 2q^{4} = 162,\ \ q^{4} = 81 \Longrightarrow q = \pm 3;\]
\[1)\ q = 3,\ \ x_{1} = 2,\ \ x_{2} = x_{1} \cdot q = 2 \cdot 3 = 6;\]
\[x_{3} = x_{1} \cdot q^{2} = 2 \cdot 3^{2} = 2 \cdot 9 = 18;\]
\[x_{4} = x_{1} \cdot q³ = 2 \cdot 3³ = 54;\ \ \ \ x_{5} = 162.\]
\[\boxed{\text{634.}\text{\ }\text{ОК\ ГДЗ\ -\ домашка\ на\ 5}}\]
\[Формула\ верна\ при\ n = 2:\ \ \]
\[\ k = n - 1 = 1;\]
\[x_{1 + 1} = - 5 + 10 \cdot 1 + 5 = 10,\ \]
\[x_{2} = 5 \cdot 2^{2} - 10 = 10.\]
\[Допустим,\ что\ при\ n = k,\ \]
\[формула\ тоже\ верна:\]
\[x_{n} = x_{k} = 5k² - 10.\]
\[Докажем,\ что\ формула\ \]
\[справедлива\ для\ n = k + 1:\]
\[x_{n} = x_{k + 1} = 5 \cdot (k + 1)^{2} - 10 =\]
\[= 5k^{2} + 10k + 5 - 10 =\]
\[= x_{k} + 10k - 5 \Longrightarrow ч.т.д.\]