Prove that $a_{n+2}=0.5(a_n+a_{n+1})$, $a_1=2, a_2=5$ has a limit, and find it I have the following sequence: $a_{n+2}=0.5(a_n+a_{n+1})$, $a_1=2, a_2=5$, and I need to prove that $\lim\limits_{n\to \infty}a_n$ exists, and find it.
I don't know ho to prove that the limit exists, since the sequence neither decreasing nor increasing.
In addition, I don't know how to find it, since the equation I am getting is $L=0.5(L+L)$, whitch is true for all $L$.

$a_1 = 2$, $a_2 = 5$
$$2a_3 = a_1 + a_2$$
$$2a_4 = a_2 + a_3$$
$$2a_5 = a_3 + a_4$$
$$\cdots$$
$$2a_{n - 1} = a_{n- 2} + a_{n - 3}$$
$$2a_n = a_{n- 1} + a_{n - 2}$$
Add all equations, you will get
$$2a_n  + a_{n - 1} = a_1 + 2a_2$$
Since you can show that the sequence is Cauchy and hence converges as I explained in the comment above, $a_n \to L$ and $a_{n - 1}\to L$ as $n \to \infty$
Hence, $$2L + L = a_1 + 2a_2$$
or $$L = \dfrac{2 + 2(5)}{3} = 4$$
