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Algebra of Derivatives

If a 1, a 2, ...

Question

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

1 < r <3

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-3 < r <-1

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r >3 or r <1

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none of these

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Solution

The correct option is C
r >3 or r <1

$a_{3} > 4 a_{2} - 3 a_{1}$

$\Rightarrow a_{1} r^{2} > 4 a_{1} r - 3 a_{1}$

$\Rightarrow r^{2} > 4 r - 3$

$\Rightarrow r^{2} - 4 r + 3 > 0$

$\Rightarrow (r - 1) (r - 3) > 0$

$\Rightarrow r < 1 o r r > 3$

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If a1,a2, a3(a1>0) are three successive terms of a G.P. with common ratio r, the value of r for which a3>4a2−3a1 holds is given by

The correct option is C r >3 or r <1 a3>4a2−3a1 ⇒a1r2>4a1r−3a1 ⇒r2>4r−3 ⇒r2−4r+3>0 ⇒(r−1)(r−3)>0 ⇒r<1 or r>3

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

The correct option is C
r >3 or r <1

$a_{3} > 4 a_{2} - 3 a_{1}$

$\Rightarrow a_{1} r^{2} > 4 a_{1} r - 3 a_{1}$

$\Rightarrow r^{2} > 4 r - 3$

$\Rightarrow r^{2} - 4 r + 3 > 0$

$\Rightarrow (r - 1) (r - 3) > 0$

$\Rightarrow r < 1 o r r > 3$