Let a1,a2,a3 be three positive numbers which are in geometric progression with common ratio r . The inequality a3a2+2a1 holds true if r is equal to

Question

Let a1,a2,a3 be three positive numbers which are in geometric progression with common ratio r . The inequality a3>a2+2a1 holds true if r is equal to (A) 2 (B) 1.5 (C) 0.5 (D) 2.5

Tardigrade Admin · Accepted Answer

a2=a1r,a3=a1r2 a3>a2+2a1 ⇒ a1r2>a1r+2a1 ⇒ r2-r-2>0 ⇒ (r - 2)(r + 1)>0 ⇒ r < -1 or r>2

Q. Let $a_{1}, a_{2}, a_{3}$ be three positive numbers which are in geometric progression with common ratio $r$ . The inequality $a_{3} > a_{2} + 2 a_{1}$ holds true if $r$ is equal to

$2$

$1.5$

$0.5$

$2.5$

$a_{2} = a_{1} r, a_{3} = a_{1} r^{2}$
$a_{3} > a_{2} + 2 a_{1}$
$\Rightarrow a_{1} r^{2} > a_{1} r + 2 a_{1}$
$\Rightarrow r^{2} - r - 2 > 0$
$\Rightarrow (r - 2) (r + 1) > 0$
$\Rightarrow r < - 1$ or $r > 2$

Q. Let a1​,a2​,a3​ be three positive numbers which are in geometric progression with common ratio r . The inequality a3​>a2​+2a1​ holds true if r is equal to

2

1.5

0.5

2.5

a2​=a1​r,a3​=a1​r2 a3​>a2​+2a1​ ⇒a1​r2>a1​r+2a1​ ⇒r2−r−2>0 ⇒(r−2)(r+1)>0 ⇒r<−1 or r>2

Q. Let $a_{1}, a_{2}, a_{3}$ be three positive numbers which are in geometric progression with common ratio $r$ . The inequality $a_{3} > a_{2} + 2 a_{1}$ holds true if $r$ is equal to

$2$

$1.5$

$0.5$

$2.5$

$a_{2} = a_{1} r, a_{3} = a_{1} r^{2}$
$a_{3} > a_{2} + 2 a_{1}$
$\Rightarrow a_{1} r^{2} > a_{1} r + 2 a_{1}$
$\Rightarrow r^{2} - r - 2 > 0$
$\Rightarrow (r - 2) (r + 1) > 0$
$\Rightarrow r < - 1$ or $r > 2$