Caught… Discover options of recursive equations utilizing genereting fucntions.

$$

x_{n+2} = 14x_{n+1} – 49x_n + n7^n

n>=0

x_0 = 1

x_2=14

$$

What I attempted :

$$

x_{n+2} = 14x_{n+1} – 49x_n + n7^n , n>=0

x_0 = 1 ; x_2=14

x_{n} = 14x_{n-1} – 49x_{n-2} + (n-2)7^{n-2} , n>=2

a_{n} = 14a_{n-1} – 49a_{n-2} + (n-2)7^{n-2}

F(x) = sum_{n=0}a_nx^n = 1 + sum_{n=1}a_nx^n

F(x) – 1 = sum_{n=1}a_nx^n

F(x) = 1 + 14x + sum_{n=2}^infty (14a_{n-1} – 49a_{n-2} + (n-2)*7^{n-2})x^n=

= 1 + 14x + 14sum_{n=2}^infty a_{n-1}x^n -49sum_{n=2}^infty a_{n-2}x^n + sum_{n=2}^infty (n-2)*7^{n-2}x^n =

= 1 + 14x + 14xsum_{n=2}^infty a_{n-1}x^{n-1} -49x^2sum_{n=2}^infty a_{n-2}x^{n-2} + x^2sum_{n=2}^infty (n-2)*7^{n-2}x^{n-2} =

= 1 + 14x + 14xsum_{n=1}^infty a_{n}x^{n} -49x^2sum_{n=0}^infty a_{n}x^{n} + x^2sum_{n=0}^infty n7^{n}x^{n} =

= 1 + 14x + 14x(F(x) – 1) – 49x^2F(x) + x^2sum_{n=0}^infty n7^{n}x^{n}

$$

And there’s a drawback. Is right? If sure how remodel

$

x^2sum_{n=0}^infty n7^{n}x^{n}

$

into one thing like $x^2F(x)$.

It is methodology which your trainer present us to get producing operate, however he did not present us how make it with inhomogeneous.