Suppose $a_0, a_1, a_2, a_3, \ldots$ is an arithmetic progression with $a_0$ and $a_1$ positive integers. Let $g_0, g_1, g_2, g_3, \ldots$ be the geometric progression such that $g_0 = a_0$ and $g_1 = a_1$. Statements (1) We must have $\left(a_5\right)^2 \geq a_0 a_{10}$. (2) The sum $a_0 + a_1 + \cdots + a_{10}$ must be a multiple of the integer $a_5$. (3) If $\sum_{i=0}^{\infty} a_i$ is $+\infty$ then $\sum_{i=0}^{\infty} g_i$ is also $+\infty$. (4) If $\sum_{i=0}^{\infty} g_i$ is finite then $\sum_{i=0}^{\infty} a_i$ is $-\infty$.
Suppose $a_0, a_1, a_2, a_3, \ldots$ is an arithmetic progression with $a_0$ and $a_1$ positive integers. Let $g_0, g_1, g_2, g_3, \ldots$ be the geometric progression such that $g_0 = a_0$ and $g_1 = a_1$.
Statements
(1) We must have $\left(a_5\right)^2 \geq a_0 a_{10}$.\\
(2) The sum $a_0 + a_1 + \cdots + a_{10}$ must be a multiple of the integer $a_5$.\\
(3) If $\sum_{i=0}^{\infty} a_i$ is $+\infty$ then $\sum_{i=0}^{\infty} g_i$ is also $+\infty$.\\
(4) If $\sum_{i=0}^{\infty} g_i$ is finite then $\sum_{i=0}^{\infty} a_i$ is $-\infty$.