Characteristic polynomial index coefficient

Question

My questions stems from the first link of this search, a downloading pdf, problem 13 of the first "Mock AIME": thomas mildorf aime.

We have a recurrence relation $64=7R_n+2R{n-1}+9R_{n-2}$, and part of the problem involves finding a closed form of the recurrence. We have $7x^3+2x^2-9x=7x^2+2x-9 \Longleftrightarrow 7x^3-5x^2-11x+9=(x-1)^2(7x+9)=0.$ Then the solution states,

This implies that we have $R_n=a\cdot n + b + c\cdot\left(\frac{-9}{7}\right)^n.$ We solve for $a,b,c$ by checking the first three terms: \begin{align}&R_0=b+c=10\\&R_1=a+b-\frac{9c}{7}=-2\\&R_2=2a+b+\frac{81c}{49}=\frac{158}{7}\end{align}

But why do we have $a\cdot n$ instead of $a$? There is a double root of $1$ in the characteristic polynomial, so why don't we have $R_n=a(1)^n+b(1)^n+c(\frac{-9}{7})^n=a+b-\frac{9}{7}c$ instead?

Does this answer your question? Homogeneous linear difference equation with multiple roots — John Omielan, Aug 20 '20 at 19:14
FYI, other related posts here include Solving for the closed term solution of a third order recurrence relation with real constant coefficients, Linear recurrence by characteristic equation., Finding the closed form solution of a third order recurrence relation with constant coefficients, Solving a recurrence relation with the characteristic polynomial, (cont.) — John Omielan, Aug 20 '20 at 19:17
... How to find unique solutions to recurrence relation $a_n =10a_{n-1}-32a_{n-2}+32a_{n-3}$ with $a_0=5$, $a_1=18$, $a_2=76$ and Show a solution to a recurrence relation?. — John Omielan, Aug 20 '20 at 19:18

Characteristic polynomial index coefficient

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