If (2x - 1) , (4x - 1), (7 + 2x) ... are in G.P, then next term of the sequence is

\frac{625}{3}

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\frac{125}{3}

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81

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9

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Solution

The correct option is A $\frac{125}{3}$
Given that (2x-1),(4x-1) and (7+2x) are in G.P
In that case, the ratio between any two consecutive terms
must be a constant
$\Rightarrow \frac{4 x - 1}{2 x - 1} = \frac{7 + 2 x}{4 x - 1}$
$\Rightarrow (4 x - 1)^{2} = (7 + 2 x) (2 x - 1)$
$\Rightarrow 16 x^{2} - 8 x + 1 = 4 x^{2} + 12 x - 7$
$\Rightarrow 12 x^{2} - 20 x + 8 = 0$
Dividing the above equation by 4 on both sides of the equality we get
$\Rightarrow 3 x^{2} - 5 x + 2 = 0$
$\Rightarrow 3 x^{2} - 3 x - 2 x + 2 = 0$
$\Rightarrow 3 x (x - 1) - 2 (x - 1) = 0$
$∴ x = 2 / 3$ or $x = 1$
For $x = 1 / 3$
the three terms will be : $\frac{1}{3}, \frac{5}{3}, \frac{25}{3}, . . . .$
the common ratio of the above GP is 5
$∴$ the next term i.e. 4th term will be : $\frac{25 \times 5}{3} = 125 / 3$
For x=1
the three terms will be : 1,3,9,...
the common ratio is 3
$∴$ the next term i.e. 4th term will be $9 \times 3 = 27$
Looking the options,
correct answer : Option (B) $[\frac{125}{3}]$

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