How to calculate the value of a coin?

Question

I do not need a specific answer, but a global formula that solves this equation.

There is a coin that costs 100\$ (or 5\$ or \$25 etc.) Each year, it steadily increases by 10% (or 1% or 4% etc.) How much did the coin cost 3 years ago?

X = price

Y = percent of increases

t = time

Thank you for help.

Answer 1

Let's say the coin costs $x_0\$$ at present. And its value increases monotonically by y% every year. Let the cost of the coin t years before was $x_{-t}\$$. Therefore,

\begin{align} &\implies x_{-t} \left(\frac{y}{100}+1\right)^t=x_0\\ &\implies x_{-t}= x_0\left(\frac{y}{100}+1\right)^{-t} \end{align}

Here, $x_{-t}\frac{y}{100}+x_{-t}$ is the increase in value $'t'$ years before.

Answer 2

See the table: $$\begin{array}{c|c|c|c|c|c} t&-2&-1&0&1&\\ \hline x_t&100(1+0.1)^{-2}&100(1+0.1)^{-1}&100&100(1+0.1)&\\ \hline x_t&x_{-2}=x_0(1+0.1)^{-2}&x_{-1}=x_0(1+0.1)^{-1}&x_0&x_1=x_0(1+r)& \end{array}$$