If b1, b2, b3(b10) are three successive terms of a G.P. with common ratio r, then value of r for which the inequality b34 b2-3 b1, holds is given by

Question

If b1, b2, b3(b1>0) are three successive terms of a G.P. with common ratio r, then value of r for which the inequality b3>4 b2-3 b1, holds is given by (A) r ≥ 3 (B) 1 < r < 2 (C) r>3 (D) r ∈(0,1)

Tardigrade Admin · Accepted Answer

b1, b2, b3 are in G.P. ∴ b3>4 b2-3 b1 ⇒ r2>4 r-3 ⇒ r2-4 r+3>0 ⇒ (r-1)(r-3) >0 So 0< r< 1 and r>3

Q. If $b_{1}, b_{2}, b_{3} (b_{1} > 0)$ are three successive terms of a G.P. with common ratio $r$ , then value of $r$ for which the inequality $b_{3} > 4 b_{2} - 3 b_{1}$ , holds is given by

$r \geq 3$

$1 < r < 2$

$r > 3$

$r \in (0, 1)$

$b_{1}, b_{2}, b_{3}$ are in G.P.
$∴ b_{3} > 4 b_{2} - 3 b_{1}$
$\Rightarrow r^{2} > 4 r - 3$
$\Rightarrow r^{2} - 4 r + 3 > 0$
$\Rightarrow (r - 1) (r - 3) > 0$
So $0 < r < 1$ and $r > 3$

Q. If b1​,b2​,b3​(b1​>0) are three successive terms of a G.P. with common ratio r, then value of r for which the inequality b3​>4b2​−3b1​, holds is given by

r≥3

1<r<2

r>3

r∈(0,1)