Percent Down / Up curve

Question

If I create a curve where, starting from a given $Y_{0}$, each new $y$ is down of $p$% and then up of $p$% (that is, the same percentage), is it known by some name?

What could be suitable equations to represent (draw) it ?

Answer 1

Reducing value $v$ by $p \%$ gives $(1-\frac p {100})v$. Increasing value $v$ by $p\%$ gives $(1 + \frac p {100})v$.

Putting those together, we find $$Y_{n+1} = \left( \frac{100-p}{100} \right) \left( \frac{100+p}{100}\right) Y_n = \frac{10000-p^2}{10000} Y_n.$$

We can apply that formula repeatedly to find: $$\begin{align} Y_{n} &= \frac{10000-p^2}{10000} Y_{n-1} \\ &= \left(\frac{10000-p^2}{10000}\right) \left( \frac{10000-p^2}{10000} \right) Y_{n-2} \\ &= \cdots \\ &= \left(\frac{10000-p^2}{10000}\right)^n Y_0. \end{align}$$

Now that we've found a formula, we find that your sequence is a geometric progression with ratio $\frac{10000-p^2}{10000}$. If you really want to see a picture, you could use WolframAlpha. For example, here's a sample link where I used command ListPlot[Table[{n, ((100-10)/100)^Ceiling[n/2] * ((100+10)/100)^Floor[n/2]}, {n, 0, 200}]] to get this picture (with $p=10, Y_0=1$)