JEE Advanced 1998

LEVELBoard

Animated Solution for Mathematics - Sequence and Series: If $x > 1, y > 1, z > 1$ are in G.P., then $\frac{1}{1 + l n x}, \frac{1}{1 + l n y}, \frac{1}{1 + l n z}$ are in

Select Answer:

Visualized Solution

Given $x, y, z$ in Geometric Progression

We are given three numbers: $x > 1$ , $y > 1$ , and $z > 1$ .
These numbers are in Geometric Progression (G.P.).
Let's visualize them on a geometric scale where each step multiplies by a common ratio.

Mathematical Definition of G.P.

For three terms $x, y, z$ in G.P., the ratio of consecutive terms is constant.
$\frac{y}{x} = \frac{z}{y} = r$ (where $r$ is the common ratio)
This gives the fundamental relation: $y^{2} = x z$

Introducing the Logarithm Tool

We need to find the progression of terms containing $ln x, ln y, ln z$ .
The natural logarithm, $ln$ , is the perfect tool to convert multiplication into addition.
Recall the key properties: $ln (ab) = ln a + ln b$ and $ln (a^{b}) = b ln a$ .

Applying Logarithms to the G.P. Relation

Take the natural logarithm (base $e$ ) on both sides of our G.P. equation:
$ln (y^{2}) = ln (x z)$
This sets up the transition from a geometric scale to an arithmetic scale.

Simplifying the Left-Hand Side

Apply the power rule of logarithms: $ln (a^{b}) = b ln a$
The left-hand side becomes:
$ln (y^{2}) = 2 ln y$

Simplifying the Right-Hand Side

Apply the product rule of logarithms: $ln (ab) = ln a + ln b$
The right-hand side becomes:
$ln (x z) = ln x + ln z$
Combining both sides: $2 ln y = ln x + ln z$

Identifying the Arithmetic Progression (A.P.)

An Arithmetic Progression (A.P.) satisfies: $2 b = a + c$
Since $2 ln y = ln x + ln z$ , the terms are in A.P.
Therefore, $ln x, ln y, ln z$ are in A.P.

Adding a Constant to the A.P.

Property of A.P.: If $a, b, c$ are in A.P., then $a + k, b + k, c + k$ are also in A.P.
Let's add $1$ to each term:
$1 + ln x, 1 + ln y, 1 + ln z$ are also in A.P.

Connecting A.P. to Harmonic Progression (H.P.)

Definition of H.P.: If $A, B, C$ are in A.P., then $\frac{1}{A}, \frac{1}{B}, \frac{1}{C}$ are in H.P.
Since $1 + ln x, 1 + ln y, 1 + ln z$ are in A.P.:
Their reciprocals $\frac{1}{1 + l n x}, \frac{1}{1 + l n y}, \frac{1}{1 + l n z}$ must be in H.P.

Final Conclusion

Taking the logarithm of a G.P. converts it into an A.P.
Adding a constant to an A.P. keeps it in A.P.
Taking the reciprocal of an A.P. converts it into an H.P.
Correct Option: B (H.P.)

00:00 / 00:00

The Sigma Insight: Harmonic Progression (H.P.)

Solution Diagram

Analyzing the Geometric Foundation

We begin with three numbers,

x, y, z > 1

, which are in a Geometric Progression (G.P.). By the definition of a G.P., the ratio between consecutive terms is constant, such that:

\frac{y}{x} = \frac{z}{y} = r

Cross-multiplying these terms yields the fundamental property of a G.P.:

y^{2} = x z

This equation serves as our anchor, representing the multiplicative nature of the sequence where the middle term squared balances the product of its neighbors.

The Logarithmic Transformation

To bridge the gap between the multiplicative world of G.P. and the linear world of Arithmetic Progressions (A.P.), we apply the natural logarithm to our anchor equation:

ln (y^{2}) = ln (x z)

Using the logarithmic power rule and the product rule, we transform the equation into:

2 ln y = ln x + ln z

This result is the defining condition for an A.P. Thus, we have successfully mapped the sequence

x, y, z

into an A.P. defined by the terms

ln x, ln y, ln z

The Arithmetic Shift

We now consider the terms

1 + ln x, 1 + ln y, 1 + ln z

. Since

ln x, ln y, ln z

are in A.P., adding a constant value of

1

to each term simply shifts the entire sequence vertically.

Because the common difference between consecutive terms remains unchanged, the sequence

1 + ln x, 1 + ln y, 1 + ln z

remains in an Arithmetic Progression.

The Harmonic Conclusion

By definition, a sequence is in a Harmonic Progression (H.P.) if the reciprocals of its terms form an Arithmetic Progression. We have established that the sequence

1 + ln x, 1 + ln y, 1 + ln z

is in A.P.

Therefore, it follows logically that their reciprocals must form an H.P.:

\frac{1}{1 + ln x}, \frac{1}{1 + ln y}, \frac{1}{1 + ln z}

Conclusion: The given terms are in Harmonic Progression (H.P.).

Common Traps & FAQs

Wait, why does taking the logarithm of a G.P. result in an A.P.? I thought logs were just for solving exponents.

That is a brilliant observation! Logarithms are indeed the bridge between the multiplicative world and the additive world. In a G.P., terms grow by multiplication (

y^{2} = x z

). When you apply the logarithm, the power rule and product rule transform that multiplicative relationship into an additive one:

2 ln y = ln x + ln z

. This is the exact definition of an Arithmetic Progression, where the middle term doubled equals the sum of the neighbors.

Does adding 1 to every term of an A.P. destroy the A.P. property? It feels like it should change the common difference.

It is a common fear, but fear not! If you have an A.P. with terms

a, b, c

and a common difference

d

, then

b - a = d

and

c - b = d

. If you add a constant

k

to every term, the new terms are

(a + k), (b + k), (c + k)

. The new difference is

(b + k) - (a + k) = b - a = d

. The common difference remains unchanged! Adding a constant just shifts the sequence on the number line; it does not warp its structure.

Is there a shortcut to identify H.P. in these types of problems?

Yes, there is a very reliable 'litmus test' for Harmonic Progressions. Whenever you see a sequence of fractions, immediately check if the denominators form an A.P. If the denominators are in A.P., the entire sequence is, by definition, in H.P. Always look for that reciprocal relationship!

Animated Solution for Mathematics - Sequence and Series: If $x > 1, y > 1, z > 1$ are in G.P., then $\frac{1}{1 + l n x}, \frac{1}{1 + l n y}, \frac{1}{1 + l n z}$ are in

Select Answer:

Visualized Solution

Given $x, y, z$ in Geometric Progression

Mathematical Definition of G.P.

Introducing the Logarithm Tool

Applying Logarithms to the G.P. Relation

Simplifying the Left-Hand Side

Simplifying the Right-Hand Side

Identifying the Arithmetic Progression (A.P.)

Adding a Constant to the A.P.

Connecting A.P. to Harmonic Progression (H.P.)

Final Conclusion

The Sigma Insight: Harmonic Progression (H.P.)

Analyzing the Geometric Foundation

The Logarithmic Transformation

The Arithmetic Shift

The Harmonic Conclusion

Similar Questions

If $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}$ where $a, b, c$ are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ then $x, y, z$ are in

Let the positive numbers $a, b, c, d$ be in A.P. Then $ab c, ab d, a c d, b c d$ are

If $a_{1}, a_{2}, \dots, a_{n}$ are in H.P., then the expression $a_{1} a_{2} + a_{2} a_{3} + \dots + a_{n - 1} a_{n}$ is equal to

Let $\frac{1}{x _{1}}, \frac{1}{x _{2}}, \dots, \frac{1}{x _{n}}$ ( $x_{i} \neq = 0$ for $i = 1, 2, \dots, n$ ) be in A.P. such that $x_{1} = 4$ and $x_{21} = 20$ . If n is the least positive integer for which $x_{n} > 50$ , then $\sum_{i = 1}^{n} (\frac{1}{x _{i}})$ is equal to :

Let $a_{1}, a_{2}, \dots, a_{10}$ be in A.P. and $h_{1}, h_{2}, \dots, h_{10}$ be in H.P. If $a_{1} = h_{1} = 2$ and $a_{10} = h_{10} = 3$ , then $a_{4} h_{7}$ is

Let $R_{1}$ and $R_{2}$ respectively be the maximum ranges up and down an inclined plane and $R$ be the maximum range on the horizontal plane. Then $R_{1}, R, R_{2}$ are in

Let $a_{1}, a_{2}, a_{3} \dots$ be in harmonic progression with $a_{1} = 5$ and $a_{20} = 25$ . The least positive integer $n$ for which $a_{n} < 0$ is

Animated Solution for Mathematics - Sequence and Series: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x>1,y>1,z>1 are in G.P., then 1+lnx1​,1+lny1​,1+lnz1​ are in

Select Answer:

Visualized Solution

Given .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x,y,z in Geometric Progression

Mathematical Definition of G.P.

Introducing the Logarithm Tool

Applying Logarithms to the G.P. Relation

Simplifying the Left-Hand Side

Simplifying the Right-Hand Side

Identifying the Arithmetic Progression (A.P.)

Adding a Constant to the A.P.

Connecting A.P. to Harmonic Progression (H.P.)

Final Conclusion

The Sigma Insight: Harmonic Progression (H.P.)

Analyzing the Geometric Foundation

The Logarithmic Transformation

The Arithmetic Shift

The Harmonic Conclusion

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=∑n=0∞​an,y=∑n=0∞​bn,z=∑n=0∞​cn where a,b,c are in A.P and ∣a∣<1,∣b∣<1,∣c∣<1 then x,y,z are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the positive numbers a,b,c,d be in A.P. Then abc,abd,acd,bcd are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a1​,a2​,…,an​ are in H.P., then the expression a1​a2​+a2​a3​+⋯+an−1​an​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​,a2​,…,a10​ be in A.P. and h1​,h2​,…,h10​ be in H.P. If a1​=h1​=2 and a10​=h10​=3, then a4​h7​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R1​ and R2​ respectively be the maximum ranges up and down an inclined plane and R be the maximum range on the horizontal plane. Then R1​,R,R2​ are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​,a2​,a3​… be in harmonic progression with a1​=5 and a20​=25. The least positive integer n for which an​<0 is

Animated Solution for Mathematics - Sequence and Series: If $x > 1, y > 1, z > 1$ are in G.P., then $\frac{1}{1 + l n x}, \frac{1}{1 + l n y}, \frac{1}{1 + l n z}$ are in

Given $x, y, z$ in Geometric Progression

If $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}$ where $a, b, c$ are in A.P and $∣ a ∣ < 1, ∣ b ∣ < 1, ∣ c ∣ < 1$ then $x, y, z$ are in

Let the positive numbers $a, b, c, d$ be in A.P. Then $ab c, ab d, a c d, b c d$ are

If $a_{1}, a_{2}, \dots, a_{n}$ are in H.P., then the expression $a_{1} a_{2} + a_{2} a_{3} + \dots + a_{n - 1} a_{n}$ is equal to

Let $a_{1}, a_{2}, \dots, a_{10}$ be in A.P. and $h_{1}, h_{2}, \dots, h_{10}$ be in H.P. If $a_{1} = h_{1} = 2$ and $a_{10} = h_{10} = 3$ , then $a_{4} h_{7}$ is

Let $R_{1}$ and $R_{2}$ respectively be the maximum ranges up and down an inclined plane and $R$ be the maximum range on the horizontal plane. Then $R_{1}, R, R_{2}$ are in

Let $a_{1}, a_{2}, a_{3} \dots$ be in harmonic progression with $a_{1} = 5$ and $a_{20} = 25$ . The least positive integer $n$ for which $a_{n} < 0$ is