Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1 + a2, b3 = b2

Q: Why can't I just assume $r=2$ and check?

Testing with a specific value is a great way to gain intuition, but in JEE Advanced, you must prove it for all valid $r$. The algebra shows that the equality holds only for impossible values of $r$.

Q: Is there a faster way to check H.P.?

You could use the harmonic mean formula, but substituting the expressions directly into the reciprocal condition is often less prone to error.

Q: Why is $r=1$ excluded?

The problem states the numbers are distinct. If $r=1$, all terms would be equal, which contradicts the 'distinct' condition.

Question

Suppose four distinct positive numbers $a_1, a_2, a_3, a_4$ are in G.P. Let $b_1 = a_1, b_2 = b_1 + a_2, b_3 = b_2 + a_3$ and $b_4 = b_3 + a_4$. STATEMENT - 1 : The numbers $b_1, b_2, b_3, b_4$ are neither in A.P. nor in G.P. and STATEMENT - 2 : The numbers $b_1, b_2, b_3, b_4$ are in H.P.

Options:

(A) STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1

(B) STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1

(C) STATEMENT - 1 is True, STATEMENT - 2 is False

(D) STATEMENT - 1 is False, STATEMENT - 2 is True

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a geometric progression (G.P.) with distinct positive terms: $a_1, a_2, a_3, a_4$. Let the first term be $a$ and the common ratio be $r$.

Given the constraints that terms are distinct and positive, we have $a > 0$, $r > 0$, and $r 
eq 1$. The terms are defined as:
$a_1 = a$, $a_2 = ar$, $a_3 = ar^2$, and $a_4 = ar^3$.

We construct a new sequence $b_n$ using the partial sums of this G.P.:
$b_1 = a$
$b_2 = a(1+r)$
$b_3 = a(1+r+r^2)$
$b_4 = a(1+r+r^2+r^3)$

### The Arithmetic Progression Test

For a sequence to be in Arithmetic Progression (A.P.), the difference between consecutive terms must be constant. We check if $b_2 - b_1 = b_3 - b_2$.

Calculating the differences:
$b_2 - b_1 = a(1+r) - a = ar$
$b_3 - b_2 = a(1+r+r^2) - a(1+r) = ar^2$

For the sequence to be in A.P., we require $ar = ar^2$. Since $a > 0$ and $r 
eq 1$, we divide by $ar$ to obtain $1 = r$.

This contradicts our initial constraint that $r 
eq 1$. Therefore, the sequence is **definitively not in A.P.**

### The Geometric Progression Test

For a sequence to be in Geometric Progression (G.P.), the ratio of consecutive terms must be constant. We check if $\frac{b_2}{b_1} = \frac{b_3}{b_2}$.

Substituting our expressions:
$$\frac{b_2}{b_1} = \frac{a(1+r)}{a} = 1+r$$
$$\frac{b_3}{b_2} = \frac{a(1+r+r^2)}{a(1+r)} = \frac{1+r+r^2}{1+r}$$

Equating these ratios:
$$(1+r) = \frac{1+r+r^2}{1+r}$$

Cross-multiplying yields $(1+r)^2 = 1+r+r^2$. Expanding the left side gives $1 + 2r + r^2 = 1 + r + r^2$.

Subtracting $1 + r + r^2$ from both sides results in $r = 0$. Since our terms are positive, $r$ must be greater than zero, meaning the sequence is **not in G.P.**

### The Harmonic Progression Test

For $b_1, b_2, b_3$ to be in Harmonic Progression (H.P.), their reciprocals must be in A.P., satisfying the condition:
$$\frac{2}{b_2} = \frac{1}{b_1} + \frac{1}{b_3}$$

Substituting our expressions and canceling $a$:
$$\frac{2}{1+r} = 1 + \frac{1}{1+r+r^2}$$

Combining the right side:
$$\frac{2}{1+r} = \frac{1+r+r^2+1}{1+r+r^2} = \frac{2+r+r^2}{1+r+r^2}$$

Cross-multiplying gives:
$$2(1+r+r^2) = (2+r+r^2)(1+r)$$
$$2 + 2r + 2r^2 = 2 + 3r + 2r^2 + r^3$$

Simplifying the equation leads to $r + r^3 = 0$, or $r(1+r^2) = 0$. Since $r > 0$, this equation has no real solution.

Thus, the sequence is **not in H.P.** We have systematically proven that the sequence $b_n$ does not follow the standard progression rules of A.P., G.P., or H.P.

Select Answer:

Visualized Solution

Visualizing the Sequences $a_{n}$ and $b_{n}$

General Representation of G.P. Terms

Expressing $b_{n}$ in terms of $a$ and $r$

Testing the A.P. Condition

Evaluating the Differences

Testing the G.P. Condition

Analyzing the G.P. Ratios

Solving for $r$ in the G.P. Equation

Testing the H.P. Condition

Algebraic Simplification of the H.P. Equation

Cross-Multiplying and Expanding

Finding the Roots for $r$

Verifying Statement 1 and Statement 2

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

Analyzing the Setup

The Arithmetic Progression Test

The Geometric Progression Test

The Harmonic Progression Test

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