جملات هندسی: $a,ar,a{r^2}$

جملات حسابی: $4a,8ar,16a{r^2}\xrightarrow{{2b = a + c}}4a + 16a{r^2} = 2(8ar)\xrightarrow{{ \div 4a}}1 + 4{r^2} = 4r$

$4{r^2} - 4r + 1 = 0 \Rightarrow {(2r - 1)^2} = 0 \Rightarrow r = \frac{1}{2}$

${a^2} + {a^2}{r^2} + {a^2}{r^4} = 4a + 8ar + 16a{r^2}\xrightarrow{{r = \frac{1}{2}}}{a^2} + \frac{{{a^2}}}{4} + \frac{{{a^2}}}{{16}} = \underbrace {4a + 4a + 4a}_{12a}$

$\xrightarrow{{ \times 16}}{a^2}(16 + 4 + 1) = 12 \times 16a \to a(21a - 12 \times 16) = 0\left\{ \begin{gathered}
  a = 0 \hfill \\
  a = \frac{{12 \times 16}}{{21}} \to a = \frac{{64}}{7} \hfill \\ 
\end{gathered}  \right.$