The hidden connection between arithmetic and geometric sequences
If we stack circles on the function $y=|x|^\color{red}{1}$, the sequence of radii is geometric. (proof)
If we stack circles on the function $y=|x|^\color{red}{2}$, the sequence of radii is arthmetic. (proof)
So if you want to know what is exactly between an arithmetic and a geometric sequence, just consider a stack of circles on the function $y=|x|^\color{red}{1.5}$.
Call the sequence of their radii $\{r_n\}$. It turns out that as $r_1\to\infty$, $r_n$ approaches the $n^\text{th}$ term of a quadratic sequence, as I show below. (Most school students will not be able to understand the explanation, but they can at least understand the result.)
From the graph, we can see that as $\frac{r_2}{r_1}\to 1$, i.e. as the gradient of the curve approches infinity,
${r_1}+{r_2}=c_2-c_1\approx {t_2}^{1.5}-{t_1}^{1.5}\approx {r_2}^{1.5}-{r_1}^{1.5}$
$\therefore\lim\limits_{\frac{r_2}{r_1}\to 1}\frac{r_1+r_2}{{r_2}^{1.5}-{r_1}^{1.5}}=1$
$\begin{align}
\lim\limits_{\frac{r_2}{r_1}\to 1}(\sqrt{r_2}-\sqrt{r_1})&=\lim\limits_{\frac{r_2}{r_1}\to 1}(\sqrt{r_2}-\sqrt{r_1})\left(\frac{r_1+r_2}{{r_2}^{1.5}-{r_1}^{1.5}}\right)\text{ using the previous result}\\
&=\lim\limits_{\frac{r_2}{r_1}\to 1}\left(\frac{r_1+r_2}{r_1}\right)\left(\frac{r_1\sqrt{r_2}-r_1\sqrt{r_1}}{{r_2}^{1.5}-{r_1}^{1.5}}\right)\text{ by rearranging}\\
&=2\lim\limits_{\frac{r_2}{r_1}\to 1}\frac{\left(\frac{r_2}{r_1}\right)^{0.5}-1}{\left(\frac{r_2}{r_1}\right)^{1.5}-1}\text{ by dividing top and bottom by ${r_1}^{1.5}$}\\
&=2\lim\limits_{\frac{r_2}{r_1}\to 1}\frac{0.5\left(\frac{r_2}{r_1}\right)^{-0.5}}{1.5\left(\frac{r_2}{r_1}\right)^{0.5}}\text{ by L'Hopital's rule}\\
&=\frac23\\
\end{align}$
So as $\frac{r_2}{r_1}\to 1$,
$\begin{align}
\sqrt{r_n}&\approx\frac23+\sqrt{r_{n-1}}\\
&\approx\frac23+\frac23+\sqrt{r_{n-2}}\\
&\approx\frac23+\frac23+\frac23+\sqrt{r_{n-3}}\\
&\approx\dots\\
&\approx\frac23(n-1)+\sqrt{r_1}\\
\end{align}$
$\therefore r_n\approx \left(\frac23(n-1)+\sqrt{r_1}\right)^2$
That is, as $r_1\to\infty$, $r_n$ approaches the $n^\text{th}$ term of a quadratic sequence.
The three most commonly taught sequences in school are the arithmetic, geometric and quadratic sequences, but most students and teachers do not realize that they have this neat relationship.