Question

Choose the correct option provided below for the following question.
If $\left( {2x - 1} \right),\left( {4x - 1} \right),\left( {7 + 2x} \right)...$ are in G.P, then next term of the sequence is
A. $\dfrac{{625}}{3}$
B. $\dfrac{{125}}{3}$
C. $81$
D. $9$

Answer 1

First, we find the ratio between the given terms because they are in geometric progression. The ratio is a constant, so it remains the same between any two terms of the progression, i.e. the ratio is a constant. From that we find the value of $x$, using the solving of quadratic equations. Now taking the two roots of $x$, we find the common ratio and using that common ratio we find the next term in the sequence.

Complete step-by-step solution:
Given, $\left( {2x - 1} \right),\left( {4x - 1} \right),\left( {7 + 2x} \right)...$ are in geometric progression.
Let us take the ratio of the terms given above and equate them because the ratio between any two terms remains constant.
$\Rightarrow \dfrac{{4x - 1}}{{2x - 1}} = \dfrac{{7 + 2x}}{{4x - 1}}$
Cross-multiplying the denominators, we get:
$\Rightarrow {\left( {4x - 1} \right)^2} = \left( {7 + 2x} \right)\left( {2x - 1} \right)$
Expanding the above terms, we get;
$\Rightarrow 16{x^2} - 8x + 1 = 4{x^2} + 12x - 7$
Simplifying the above equation after rearranging, we get;
$\Rightarrow 12{x^2} - 20x + 8 = 0$
Taking the common term out and then dividing the equation with that term on both the sides, we get:
$\Rightarrow 3{x^2} - 5x + 2 = 0$
Using the factorisation method, we can write the above equation as;
$\Rightarrow 3{x^2} - 3x - 2x + 2 = 0$
Taking out the common term, we get;
$\Rightarrow 3x(x - 1) - 2(x - 1) = 0$
Getting the common terms out yet again, we get;
$\Rightarrow (x - 1)(3x - 2) = 0$
Now to find the roots of $x$, we need to take the above equation and equate one at a time to zero.
$\Rightarrow x = \dfrac{2}{3}$ or $x = 1$
Now, we have two values for $x$.
Taking each value once at a time, we find the common ratio and the fourth term.
Let us consider the first term to be $a$
Common ratio be $r$
Substituting the value of $x$ in the given question, we get the first three terms;
For $x = \dfrac{2}{3}$
$\left( {2x - 1} \right),\left( {4x - 1} \right),\left( {7 + 2x} \right)...$
$\Rightarrow \left( {2\left( {\dfrac{2}{3}} \right) - 1} \right),\left( {4\left( {\dfrac{2}{3}} \right) - 1} \right),\left( {7 + \left( {\dfrac{2}{3}} \right)} \right)...$
$\Rightarrow \left( {\dfrac{4}{3} - 1} \right),\left( {\dfrac{8}{3} - 1} \right),\left( {7 + \dfrac{2}{3}} \right)...$
Simplifying the equation;
$\Rightarrow \dfrac{1}{3},\dfrac{5}{3},\dfrac{{25}}{3}...$
The common ratio of the series = \dfrac{5}{3} \times \dfrac{3}{1}$$$$ = 5
The fourth term in the series = \dfrac{{25}}{3} \times 5$$$$ = \dfrac{{125}}{3}
Therefore, the next term is $\dfrac{{125}}{3}$
For $x = 1$
$\left( {2x - 1} \right),\left( {4x - 1} \right),\left( {7 + 2x} \right)...$
$\Rightarrow \left( {2\left( 1 \right) - 1} \right),\left( {4\left( 1 \right) - 1} \right),\left( {7 + 2\left( 1 \right)} \right)...$
Simplifying the above terms, we get;
$\Rightarrow 1,3,9...$
The common ratio in the series $= \dfrac{3}{1} = 3$
The fourth term of the series $= 9 \times 3 = 27$
Hence, the next term in the series is $27$.
Looking at the options given, the first case, i.e. for $x = \dfrac{2}{3}$, the term is $\dfrac{{125}}{3}$.

$\therefore$ The correct option is B.

Note: We have to mind that, in mathematics a geometric progression also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed. Non-one number called the common ratio. Examples of a geometric sequence are powers ${r^k}$ of a fixed number $r$, such as ${2^k}$ and ${3^k}$. The general form of a geometric sequence is,
$a,{\text{ }}ar,{\text{ }}a{r^2},{\text{ }}a{r^3},{\text{ }}a{r^4},{\text{ }}........$ Where $r \ne 1$ is the common ratio and $a$ is a scale factor, equal to the sequence’s start value.