How do I turn the equation of a set into the equation of a line?

Question

The equation of my set is: X(i) = 1.1 * X(i-1) + 10 where X(i) = 10

I want to know how to turn this into a equation into which I hope to insert a value of X(i-50) (for example) and find X(i) or graph on a graphic calculator.

Answer 1

It looks like you are trying to solve a linear difference equation. These are of the form $$x_{n+1}=ax_n+b$$

There are two special cases:

1. If $a=1$ then $x_n=x_0+nb$
2. If $b=0$ then $x_n = a^nx_0$

To solve the general case we can transform our original equation into form $2$ by introducing a new set of variables $y_n$ and making a substitution.

For each $n$, let $y_n=x_n-c$ where $c$ is a constant we will set to suit our needs. Note that $c$ is the same for each $n$.

Then $x_{n+1}=ax_n+b$ becomes $y_{n+1}+c = a(y_n+c)+b = ay_n+ac+b$.

Here's the trick: Since we are free to set $c$ to be anything we want we will set it so that $c = ac+b\implies c=\frac{b}{1-a}$. Notice that we dealt with the special case $a=1$ above so we can assume $a\ne 1$ so we don't get a division by zero.

Now $y_{n+1}+c = ay_n+ac+b$ becomes $y_{n+1}=ay_n\implies y_n=a^ny_0$

Replacing $y_n$ with $x_n-c$ gives $$x_n-c = a^n(x_0-c)$$

Your numbers $a=1.1$, $b=10$ and $x_0 = 1$ give us that $c=\frac{10}{-.1}=-100$ and this gives us $$x_n=(1.1)^n(101) - 100$$

Answer 2

You have an inhomogeneous linear recurrence relation. You can generate a table of values in a spreadsheet by making a column for $i$ and a column for $X(i)$, writing the equation into the $X(2)$ square and copying down. You get the below. The spreadsheet will graph it for you as well. John Douma has shown an analytic solution.