Three numbers are in an increasing geometric progression with common ratio r. If the middle number i

Q: Wait, why did you choose $\frac{a}{r}, a, ar$ instead of the standard $a, ar, ar^2$?

That is a fantastic question! While both sets of variables are mathematically valid, choosing $\frac{a}{r}, a, ar$ is a strategic move. In many problems involving products or symmetric sums of three terms in G.P., this choice allows the common ratio $r$ to cancel out or simplify beautifully. It saves you time and reduces the chance of algebraic errors—a vital skill for the JEE Advanced!

Q: Why do we reject the root $r = 2 - \sqrt{3}$?

Great catch! The problem explicitly states that the Geometric Progression is **increasing**. For a G.P. to be increasing, the common ratio $r$ must be greater than $1$ (assuming the first term is positive). Since $\sqrt{3} \approx 1.732$, the value $2 - \sqrt{3} \approx 0.268$, which is clearly less than $1$. Therefore, it cannot be the common ratio for an increasing G.P.

Q: How do I know when to rationalize the denominator?

Rationalization is a standard tool in your toolkit whenever you encounter an irrational expression in the denominator, especially when you need to perform further arithmetic operations like subtraction or addition. In this problem, we had $d = 6 - \frac{3}{2 + \sqrt{3}}$. Rationalizing the fraction $\frac{3}{2 + \sqrt{3}}$ by multiplying the numerator and denominator by the conjugate $(2 - \sqrt{3})$ turns the denominator into a simple integer, making the final subtraction trivial. Always look for this when you see radicals!

Question

Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of $GP$ is $3r^2$, then $r^2 - d$ is equal to :

Options:

(A) $7 - 7\sqrt{3}$

(B) $7 + \sqrt{3}$

(C) $7 - \sqrt{3}$

(D) $7 + 3\sqrt{3}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To solve this problem, we represent the three numbers in an increasing Geometric Progression (G.P.) as:
$$\frac{a}{r}, a, ar$$

This symmetric choice simplifies the algebraic manipulation significantly. We must also note the constraint that the progression is increasing, which implies that the common ratio must satisfy $r > 1$.

### The Transformation: From Geometric to Arithmetic

The problem states that doubling the middle term transforms the sequence into an Arithmetic Progression (A.P.). The new sequence is:
$$\frac{a}{r}, 2a, ar$$

By the definition of an A.P., the middle term is the arithmetic mean of its neighbors. This leads to the following equation:
$$2(2a) = \frac{a}{r} + ar$$

### The Quadratic Battle

We simplify the equation by factoring out $a$ (given $a 
eq 0$):
$$4a = a\left(\frac{1}{r} + right)$$
$$4 = \frac{1}{r} + r$$

Multiplying by $r$ yields the quadratic equation:
$$r^2 - 4r + 1 = 0$$

Using the quadratic formula, $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we find:
$$r = \frac{4 \pm \sqrt{16 - 4}}{2} = 2 \pm \sqrt{3}$$

Given our constraint $r > 1$, we discard $2 - \sqrt{3} \approx 0.268$. Thus, the common ratio is fixed at:
$$r = 2 + \sqrt{3}$$

### The Final Pieces of the Puzzle

The problem specifies that the fourth term of the original G.P. is $3r^2$. Since the fourth term is $ar^2$, we equate:
$$ar^2 = 3r^2 \Rightarrow a = 3$$

Next, we calculate the common difference $d$ of the A.P., defined as the difference between the second and first terms:
$$d = 2a - \frac{a}{r} = 6 - \frac{3}{2 + \sqrt{3}}$$

Rationalizing the denominator by multiplying by the conjugate $(2 - \sqrt{3})$:
$$d = 6 - 3(2 - \sqrt{3}) = 6 - 6 + 3\sqrt{3} = 3\sqrt{3}$$

Finally, we compute $r^2 - d$. First, calculate $r^2$:
$$r^2 = (2 + \sqrt{3})^2 = 4 + 3 + 4\sqrt{3} = 7 + 4\sqrt{3}$$

Subtracting $d = 3\sqrt{3}$ from $r^2$:
$$r^2 - d = (7 + 4\sqrt{3}) - 3\sqrt{3} = 7 + \sqrt{3}$$

The final result is **$7 + \sqrt{3}$**.

Select Answer:

Visualized Solution

Defining the Geometric Progression

Transition to Arithmetic Progression

Applying the A.P. Property

Simplifying the Equation

Forming the Quadratic Equation

Solving for $r$

Selecting the Correct $r$

Finding the Value of $a$

Calculating Common Difference $d$

Rationalizing the Expression for $d$

Final Calculation: $r^{2} - d$

The Sigma Insight: Geometric Progression (G.P.)

Analyzing the Setup

The Transformation

The Quadratic Battle

The Final Pieces of the Puzzle

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