Consider the graph of a cubic polynomial y=x3+a x2+b x+c as shown in the figure. If roots of the cubic equation x 3+ ax 2+ bx + c =0 are α, 1, β(α, β ∈ I ) such that α, 1, β (in that order) form the first three terms of an arithmetic progression, then find its 5001 text th term.

Question

Consider the graph of a cubic polynomial y=x3+a x2+b x+c as shown in the figure. If roots of the cubic equation x 3+ ax 2+ bx + c =0 are α, 1, β(α, β ∈ I ) such that α, 1, β (in that order) form the first three terms of an arithmetic progression, then find its 5001 text th term. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

f (0)=3 gives c =3 and product of roots 1 ⋅ α ⋅ β=- c ⇒ α β=-3 Also 2=α+β as α, 1, β are in A.P. ∵ roots are integer ⇒ α=-1 and β=3 ∴ First three term =-1,1,3 (form an A.P.) ∵ a =-1, d =2 ∴ T 5001=-1+5000 × 2=9999.

Q. Consider the graph of a cubic polynomial y=x3+ax2+bx+c as shown in the figure. If roots of the cubic equation x3+ax2+bx+c=0 are α,1,β(α,β∈I) such that α,1,β (in that order) form the first three terms of an arithmetic progression, then find its 5001th term.

f(0)=3 gives c=3 and product of roots 1⋅α⋅β=−c⇒αβ=−3 Also 2=α+β as α,1,β are in A.P. ∵ roots are integer ⇒α=−1 and β=3 ∴ First three term =−1,1,3 (form an A.P.) ∵a=−1,d=2 ∴T5001​=−1+5000×2=9999.