how to get natural exponential equation with points?

Question

Here are my points

$x = 0, y = 10000 \\ x = 1, y = 10421.716 \\ x = 2, y = 10882.302 \\ x = 3, y = 11385.341\\ x = 4, y = 11934.746 \\x = 5, y = 12534.791 \\x = 6, y = 13190.142 \\ x = 7, y = 13905.899\\ x = 8, y = 14687.628$

Here are my values.

The initial value for $y$ is $10000$. As $x$ value increase by $1$, $y$ is multiplied by $1.8^{0.15}$ and subtracted by $500$.

How can I make natural exponential equation with these points? or is it impossible to make natural exponential equation with this?

Answer 1

Let's rewrite the terms so that we see the pattern emerging. For brevity, I'll set $1.8^{0.15} = q$: $$ 10~000 = 10~000 $$ $$10~000 q - 500 = 10~421.72 $$ $$ 10~000q^2 -500q^1 -500q^0 = 10~882.302$$ $$ 10~000q^3 -500q^2 -500q^1 - 500 = 11~385.34$$ $$ 10~000q^4 -500q^3 -500q^2 - 500q - 500 = 11~934.75$$ The last one we can rewrite in a bit clearer form: $$ 10~000q^4 -500(q^3+q^2+q + 1) = 11~934.75$$ We see that, when $x$ is increased by $1$, the exponent in the first term is increased by one and another power of $q$ appears in the bracketed sum. The sum in the brackets is a geometric sum, which is equal to (proof is omitted) $$ 1 + q + q^2 + q^3 + \dots + q^{n-1} = \frac{1-q^{n}}{1-q} $$ Plugging this result in, we get the general expression $$ y(x) = 10~000q^{x} - 500 \frac{1-q^x}{1-q} \qquad \text{with }q=1.8^{0.15} $$ If we want to rewrite this in terms of $e$, we can set $r = \ln{1.8^{0.15}} = 0.15\ln{1.8}$ and then $$ \begin{split} y(x) &=& 10~000e^{rx} - 500 \frac{1-e^{rx}}{1-e^r}\\ &=& 10~000\exp{(rx)} - 500 \frac{1-\exp{(rx)}}{1-\exp{(r)}}\\ \end{split} $$

Answer 2

The solution for the sequence \begin{align} y_0&=a,\quad y_{n+1}=b\,y_n+c,\quad b\ne1 \\ \text{is }\quad y_n&=b^n\left(a+\frac{c}{b-1}\right) -\frac{c}{b-1} . \end{align}

For this question you have

\begin{align} a&=10000,\quad b=1.8^{0.15},\quad c=-500 \\ \text{and }\quad y_n&\approx 4575.335040\cdot1.8^{0.15\,n}+5424.664960 . \end{align}

For the other your question you have

\begin{align} a&=0,\quad b=0.95,\quad c=5\cdot0.95=4.75 \\ \text{and }\quad y_n&= -95\cdot0.95^n+95 . \end{align}