How create an exponential equation with points? (natural exponential)

Question

How do I make exponential equation with points?

Here are my points:

x = 0, y = 0
x = 1, y = 4.75000
x = 2, y = 9.26250
x = 3, y = 13.54937
x = 4, y = 17.62191

this is same as

x = 0, y = 0
x = 1, y = (0 + 5) * 0.95 
x = 2, y = ((5 * 0.95) + 5) * 0.95
x = 3, y = (((5 * 0.95) + 5) * 0.95 + 5)) * 0.95
x = 4, y = ((((5 * 0.95) + 5) * 0.95 + 5)) * 0.95 + 5) ) 0.95

as x value increase by 1, y value is added by 5 and multiplied by 0.95 i tried to make an equation for this but i couldn't

also i tried to use natural exponential but the initial value is 0 so it didn't work

please help me

Answer 1

The recurrence is

$$y_{n+1}=0.95(y_n+5)$$ that you can rewrite as

$$y_{n+1}-95=0.95\,(y_n-95).$$

Then by induction,

$$y_n-95=0.95^n(y_0-95)=-95\cdot0.95^n$$ and

$$\color{green}{y_n=95(1-0.95^n)}.$$

Answer 2

You can rewrite this in terms of a recurrence relation: $a_n = \big(a_{n-1}+ 5 \big) \cdot 0.95$, where $a_n$ is the $n$th term.

According to Wolfram Alpha, the closed form with $a_0 = 0$ is $-95 \big ((0.95)^n - 1 \big)$, and graphing this on Desmos gives this graph.