If ( n +3)2= a ( n +2)2+ b ( n +1)2+ cn 2 holds true for every positive integer n then the quadratic equation ax 2+ bx + c =0 has

Question

If ( n +3)2= a ( n +2)2+ b ( n +1)2+ cn 2 holds true for every positive integer n then the quadratic equation ax 2+ bx + c =0 has (A) both positive roots (B) both negative roots (C) one positive and one negative root (D) no real roots.

Tardigrade Admin · Accepted Answer

a =3 ; b =-3 text and c =1 ∴ 3 x 2-3 x +1=0 D <0 ⇒ text no real roots

Q. If $(n + 3)^{2} = a (n + 2)^{2} + b (n + 1)^{2} + c n^{2}$ holds true for every positive integer $n$ then the quadratic equation $a x^{2} + b x + c = 0$ has

both positive roots

both negative roots

one positive and one negative root

no real roots.

$a = 3; b = - 3 and c = 1$
$∴ 3 x^{2} - 3 x + 1 = 0$
$D < 0 \Rightarrow no real roots$

Q. If (n+3)2=a(n+2)2+b(n+1)2+cn2 holds true for every positive integer n then the quadratic equation ax2+bx+c=0 has