# Limit of sequence in which each term is defined by the average of preceding two terms

We have a sequence of numbers $x_n$ determined by the equality

The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $x_0$ and $x_1$

The options are:

A)$\frac{x_0 + 2x_1}{3}$
B)$\frac{2x_0 + 2x_1}{3}$

C)$\frac{2x_0 + 3x_1}{3}$
D)$\frac{2x_0 - 3x_1}{3}$

Since it was a multiple choice exam I plugged $x_0=1$ and $x_1=1$. Which means that all terms of this sequence is $1$,i.e,

From this I concluded that option A was correct.I could not find any way to solve this one hence I resorted to this trick. What is the actual method to find the sequence’s limit?

Supposing that $x_n$ has a limit $L$ then making $n\to \infty$ we get: