JEE Advanced 1994

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Animated Solution for Mathematics - Sequence and Series: If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

Q: How do I know when it is safe to drop the logarithms from both sides?

You can drop the logarithms when you have the form $\ln(A) = \ln(B)$. Because the natural logarithm function $f(x) = \ln(x)$ is a one-to-one function (meaning for every output, there is exactly one input), if $\ln(A) = \ln(B)$, then $A$ must equal $B$. Just ensure you have a single log term on both sides first!

Q: What exactly is a Harmonic Progression (H.P.)?

A sequence $a, b, c$ is in Harmonic Progression if their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in Arithmetic Progression. This is equivalent to saying $b$ is the harmonic mean of $a$ and $c$, defined by the formula $b = \frac{2ac}{a+c}$. Our derivation led us exactly to this relationship, confirming the sequence is in H.P.

Select Answer:

Visualized Solution

Given Condition

We are given three logarithmic terms in Arithmetic Progression (A.P.).
Term 1: $ln (a + c)$
Term 2: $ln (a - c)$
Term 3: $ln (a - 2 b + c)$

Condition for A.P.

If three numbers $x, y, z$ are in A.P., the middle term is the arithmetic mean of the other two.
Mathematical condition: $2 y = x + z$

Applying the A.P. Condition

Let's substitute our specific logarithmic terms into the A.P. formula.
$2 ln (a - c) = ln (a + c) + ln (a - 2 b + c)$

Logarithm Power Rule

Recall the power rule for logarithms: $n ln (x) = ln (x^{n})$
Applying this to the Left Hand Side (LHS):
$2 ln (a - c) \to ln ((a - c)^{2})$

Logarithm Product Rule

Recall the product rule for logarithms: $ln (x) + ln (y) = ln (x y)$
Applying this to the Right Hand Side (RHS):
$ln (a + c) + ln (a - 2 b + c) \to ln ((a + c) (a - 2 b + c))$

Equating the Arguments

Our equation is now: $ln ((a - c)^{2}) = ln ((a + c) (a - 2 b + c))$
Since the natural logarithm is a one-to-one function, we can equate the arguments:
$(a - c)^{2} = (a + c) (a - 2 b + c)$

Expanding the Left Hand Side

Let's expand the LHS using the standard algebraic identity: $(x - y)^{2} = x^{2} - 2 x y + y^{2}$
LHS: $(a - c)^{2} = a^{2} - 2 a c + c^{2}$

Expanding the Right Hand Side

Now, expand the RHS by distributing $(a + c)$ into $(a - 2 b + c)$ :
RHS: $a (a - 2 b + c) + c (a - 2 b + c)$
RHS: $a^{2} - 2 ab + a c + a c - 2 b c + c^{2}$

Simplifying the Right Hand Side

Combine the like terms on the RHS.
Notice the two $a c$ terms: $a c + a c = 2 a c$
Simplified RHS: $a^{2} - 2 ab + 2 a c - 2 b c + c^{2}$

Equating and Canceling Terms

Bring the LHS and RHS back together:
$a^{2} - 2 a c + c^{2} = a^{2} - 2 ab + 2 a c - 2 b c + c^{2}$
Cancel $a^{2}$ and $c^{2}$ from both sides.
Result: $- 2 a c = - 2 ab + 2 a c - 2 b c$

Rearranging the Equation

Let's group the terms containing $b$ on one side.
Move $- 2 ab$ and $- 2 b c$ to the left side: $2 ab + 2 b c$
Move $- 2 a c$ to the right side: $2 a c + 2 a c = 4 a c$
Equation becomes: $2 ab + 2 b c = 4 a c$

Solving for $b$

Factor out $2 b$ on the left side: $2 b (a + c) = 4 a c$
Divide both sides by $2$ : $b (a + c) = 2 a c$
Isolate $b$ : $b = \frac{2 a c}{a + c}$

Conclusion: Harmonic Progression

We derived the relation: $b = \frac{2 a c}{a + c}$
This is the exact mathematical definition of the Harmonic Mean.
Final Answer: The terms $a, b, c$ are in H.P.

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The Sigma Insight: Relation Between A.M., G.M., and H.M.

Analyzing the Setup

The problem presents three terms:

ln (a + c)

ln (a - c)

, and

ln (a - 2 b + c)

. We are given that these terms exist in an Arithmetic Progression (A.P.).

The fundamental property of an A.P. with three terms

x, y, z

is that the middle term is the arithmetic mean of the outer terms. This is expressed by the relation:

2 y = x + z

Applying this to our specific logarithmic terms, we obtain the master equation:

2 ln (a - c) = ln (a + c) + ln (a - 2 b + c)

The Logarithmic Alchemy

To solve for the relationship between

a, b,

and

c

, we must simplify the logarithmic expression. We utilize the power rule,

n ln (x) = ln (x^{n})

, to rewrite the left side:

ln ((a - c)^{2}) = ln (a + c) + ln (a - 2 b + c)

Next, we apply the product rule,

ln (x) + ln (y) = ln (x y)

, to the right side of the equation:

ln ((a - c)^{2}) = ln ((a + c) (a - 2 b + c))

Since the natural logarithm is a one-to-one function, we can equate the arguments directly:

(a - c)^{2} = (a + c) (a - 2 b + c)

The Algebraic Expansion

We now expand both sides of the equation. The left side follows the identity

(x - y)^{2} = x^{2} - 2 x y + y^{2}

a^{2} - 2 a c + c^{2} = (a + c) (a - 2 b + c)

Expanding the right side by distributing the terms, we get:

a^{2} - 2 ab + a c + a c - 2 b c + c^{2}

Combining the like terms on the right side, the equation becomes:

a^{2} - 2 a c + c^{2} = a^{2} - 2 ab + 2 a c - 2 b c + c^{2}

The Harmonic Reveal

We observe that

a^{2}

and

c^{2}

appear on both sides and cancel out. This leaves us with:

- 2 a c = - 2 ab + 2 a c - 2 b c

Rearranging the terms to isolate the components involving

b

on one side, we get:

2 ab + 2 b c = 4 a c

Dividing the entire equation by

2

, we arrive at:

ab + b c = 2 a c

Factoring out

b

, we find:

b (a + c) = 2 a c

Solving for

b

, we reach the final result:

b = \frac{2 a c}{a + c}

This expression is the classic definition of the Harmonic Mean. Therefore, we have proven that

a, b,

and

c

are in Harmonic Progression.

Common Traps & FAQs

Wait, why can't we just say

a, b, c

are in A.P. because the logs are in A.P.?

That is a classic trap! Logarithms do not preserve the A.P. property. If

ln (x), ln (y), ln (z)

are in A.P., it implies

2 ln (y) = ln (x) + ln (z)

, which simplifies to

y^{2} = x z

. This actually means

x, y, z

are in Geometric Progression, not Arithmetic Progression. Always rely on the definition of the progression, not the form of the terms.

How do I know when it is safe to drop the logarithms from both sides?

You can drop the logarithms when you have the form

ln (A) = ln (B)

. Because the natural logarithm function

f (x) = ln (x)

is a one-to-one function (meaning for every output, there is exactly one input), if

ln (A) = ln (B)

, then

A

must equal

B

. Just ensure you have a single log term on both sides first!

What exactly is a Harmonic Progression (H.P.)?

A sequence

a, b, c

is in Harmonic Progression if their reciprocals

\frac{1}{a}, \frac{1}{b}, \frac{1}{c}

are in Arithmetic Progression. This is equivalent to saying

b

is the harmonic mean of

a

and

c

, defined by the formula

b = \frac{2 a c}{a + c}

. Our derivation led us exactly to this relationship, confirming the sequence is in H.P.

Animated Solution for Mathematics - Sequence and Series: If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

Select Answer:

Visualized Solution

Given Condition

Condition for A.P.

Applying the A.P. Condition

Logarithm Power Rule

Logarithm Product Rule

Equating the Arguments

Expanding the Left Hand Side

Expanding the Right Hand Side

Simplifying the Right Hand Side

Equating and Canceling Terms

Rearranging the Equation

Solving for $b$

Conclusion: Harmonic Progression

The Sigma Insight: Relation Between A.M., G.M., and H.M.

Analyzing the Setup

The Logarithmic Alchemy

The Algebraic Expansion

The Harmonic Reveal

Similar Questions

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

Let $f : R \to R$ be such that for all $x \in R (2^{1 + x} + 2^{1 - x}), f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :

Let $f : R \to R$ be such that for all $x \in R$ , $(2^{1 + x} + 2^{1 - x})$ , $f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ( $l, n > 1$ ) and $G_{1}, G_{2}$ and $G_{3}$ are three geometric means between $l$ and $n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals:

Comprehension Passage

Which one of the following statements is correct ?

Which one of the following statements is correct ?

Which one of the following statements is correct?

For $x \geq 0$ , the least value of $K$ , for which $4^{1 + x} + 4^{1 - x}, \frac{K}{2}, 1 6^{x} + 1 6^{- x}$ are three consecutive terms of an A.P., is equal to :

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 ln(a+c),ln(a−c),ln(a−2b+c) are in A.P., then

Select Answer:

Visualized Solution

Given Condition

Condition for A.P.

Applying the A.P. Condition

Logarithm Power Rule

Logarithm Product Rule

Equating the Arguments

Expanding the Left Hand Side

Expanding the Right Hand Side

Simplifying the Right Hand Side

Equating and Canceling Terms

Rearranging the Equation

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b

Conclusion: Harmonic Progression

The Sigma Insight: Relation Between A.M., G.M., and H.M.

Analyzing the Setup

The Logarithmic Alchemy

The Algebraic Expansion

The Harmonic Reveal

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in A.P., a2,b2,c2 are in H.P., then prove that either a=b=c or a,b,−c/2 form a G.P.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the first and the (2n−1)st terms of an A.P., a G.P. and an H.P. are equal and their nth terms are a,b and c respectively, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be such that for all x∈R(21+x+21−x),f(x) and (3x+3−x) are in A.P., then the minimum value of f(x) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be such that for all x∈R, (21+x+21−x), f(x) and (3x+3−x) are in A.P., then the minimum value of f(x) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If m is the A.M. of two distinct real numbers l and n (l,n>1) and G1​,G2​ and G3​ are three geometric means between l and n, then G14​+2G24​+G34​ equals:

Comprehension Passage

Which one of the following statements is correct ?

Which one of the following statements is correct ?

Which one of the following statements is correct?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For x≥0, the least value of K, for which 41+x+41−x,2K​,16x+16−x are three consecutive terms of an A.P., is equal to :

Animated Solution for Mathematics - Sequence and Series: If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

Solving for $b$

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

Let $f : R \to R$ be such that for all $x \in R (2^{1 + x} + 2^{1 - x}), f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :

Let $f : R \to R$ be such that for all $x \in R$ , $(2^{1 + x} + 2^{1 - x})$ , $f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ( $l, n > 1$ ) and $G_{1}, G_{2}$ and $G_{3}$ are three geometric means between $l$ and $n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals:

For $x \geq 0$ , the least value of $K$ , for which $4^{1 + x} + 4^{1 - x}, \frac{K}{2}, 1 6^{x} + 1 6^{- x}$ are three consecutive terms of an A.P., is equal to :