If $\alpha,\beta$ be the roots of $ax^2+bx+c=0 (a,b,c \in R) , \frac{c}{a}<1$ and $b^2-4ac <0$, $$f(n) \sum^n_{r=1} (|\alpha|^r +|\beta|^r)$$ then $$\lim_{n\to \infty} f(n) $$ is equal to ?
Sum of the roots $\alpha + \beta =-\frac{b}{a}, $ Prouduct of roots $\alpha \beta =\frac{c}{a}$ Now how to find the value of $\lim_{n\to \infty} f(n) $ please suggest as I don't have any clue on this thanks.
Can we use the equation $3x^2+2x+1 =0$ for this any how, as it fulfills the required condition in the problem.
Hint: try to prove $|\alpha|^2=|\beta|^2=\dfrac{c}{a}<1$, then $\lim f(n)=\dfrac{|\alpha|}{1-|\alpha|}+\dfrac{|\beta|}{1-|\beta|}$