Question

Consider the graph of a cubic polynomial $y=x^3+a x^2+b x+c$ as shown in the figure. If roots of the cubic equation $x ^3+ ax ^2+ bx + c =0$ are $\alpha, 1, \beta(\alpha, \beta \in I )$ such that $\alpha, 1, \beta$ (in that order) form the first three terms of an arithmetic progression, then find its $5001^{\text {th }}$ term.

Answer 1

$f (0)=3$ gives $c =3$ and product of roots $1 \cdot \alpha \cdot \beta=- c \Rightarrow \alpha \beta=-3$
Also $2=\alpha+\beta$ as $\alpha, 1, \beta$ are in A.P.
$\because $ roots are integer $\Rightarrow \alpha=-1$ and $\beta=3$
$\therefore $ First three term $=-1,1,3 $ (form an A.P.)
$\because a =-1, d =2$
$\therefore T _{5001}=-1+5000 \times 2=9999$.