Sorry in advance as English is not my primary language.
I randomly thought of the following simple problem, and I coudn't solve it after one one hour trying. Maybe you guys can help.
Let $a_1$ and $a_2$ be positive real numbers. Let $a_n$ be the arithmetic average of the previous 2 numbers, i.e.:
$$a_n = \frac{a_{n-1}+a_{n-2}}{2}$$
If I draw this as points on a paper, I can obviously see that the limit of $a_n$ as $n\rightarrow\infty$ is a function of $a_1$ and $a_2$, but I can't solve it.
How do I proceed?
If you rewrite the equation as
you obtain a so called homogeneous linear difference equation.
This type of equation can be solved.
A possible method is to check what happens if you plug in a "guessed" solution of the form $a_n = c\cdot \lambda^n$ where $c$ is a real constant.
You will find that the solution can be written as $$a_n = c_1\cdot 1^n + c_2\cdot \left(-\frac{1}{2} \right)^n = c_1 + c_2\cdot \left(-\frac{1}{2} \right)^n$$
The $1$ and $-\frac{1}{2}$ come from solving the quadratic equation $2\lambda^2 - \lambda - 1 = 0$ you will come across when you carry out the suggested approach $a_n = c\cdot \lambda^n$.