Question: Let us suppose $x\_1,...,x\_n$ be $n$ nodes and we interpolate the functions $f$ with lagrange polynomial 

Then my book says $$\int\_a^b f(x) dx=\sum A\_i f(x\_i), A\_i=\int\_a^b l\_i(x)dx $$where $ l\_i(x\_j) = \delta\_{ij}$

for all polynomial for degree at most n. Certainly this is true for polynomial of degree $n$, because there is unique polynomial of degree n passing through these points, but why it has to be true for polynomial of degree less than $n$?

Answer: 
Just like with linear differential equations, you have to find a particular solution of the non-homogeneous recurrence equation, and add it to the general solution of the homogeneous recurrence equation.

Now the non-homogeneous part has the standard form of an exponential times a linear polynomial. So a particular solution we're seeking for will have the form $y\_n=p(n)(-2)^n$, where $\deg p$ has degree $1$ more, because $-2$ is a simple root of the characteristic equation:
$$y\_n=(\alpha n^2+\beta n)(-2)^n$$
Can you proceed now?
Hint

WLOG $a\_m=b\_m+c\_0+c\_1m+\cdots$

Replace the values of $a\_j, j=n,n+1,n+2$ and compare the coefficients of $n^0,n^1,n^2$

to find $c\_j=0\forall j\ge2$ and the values of $c\_0,c\_1$

so that $$b\_{n+2}-5b\_{n+1}-14b\_n=0$$