Let A1, G1, H1 denote the arithmetic,...

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2007Moderate

Comprehension Passage

Let

A_{1}, G_{1}, H_{1}

denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For

n \geq 2

, Let

A_{n - 1}

and

H_{n - 1}

have arithmetic, geometric and harmonic means as

A_{n}, G_{n}, H_{n}

respectively.

Question 1:

Which one of the following statements is correct ?

(A)

G_{1} > G_{2} > G_{3} > \dots

(B)

G_{1} < G_{2} < G_{3} < \dots

(C)

G_{1} = G_{2} = G_{3} = \dots

(D)

G_{1} < G_{3} < G_{5} < \dots

and

G_{2} > G_{4} > G_{6} > \dots

Answer:C

Question 2:

Which one of the following statements is correct ?

(A)

A_{1} > A_{2} > A_{3} > \dots

(B)

A_{1} < A_{2} < A_{3} < \dots

(C)

A_{1} > A_{2} > A_{5} > \dots

and

A_{2} < A_{4} < A_{6} < \dots

(D)

A_{1} < A_{3} < A_{5} < \dots

and

A_{2} > A_{4} > A_{6} > \dots

Answer:A

Question 3:

Which one of the following statements is correct?

(A)

H_{1} > H_{2} > H_{3} > \dots

(B)

H_{1} < H_{2} < H_{3} < \dots

(C)

H_{1} > H_{3} > H_{5} > \dots

and

H_{2} < H_{4} < H_{6} < \dots

(D)

H_{1} < H_{3} < H_{5} < \dots

and

H_{2} > H_{4} > H_{6} > \dots

Answer:B

Visualized Solution (English)

Visualizing the Initial Means

Let the two distinct positive numbers be $a$ and $b$ .
Initial means are defined as:
$A_{1} = \frac{a + b}{2}$ (Arithmetic Mean)
$G_{1} = ab$ (Geometric Mean)
$H_{1} = \frac{2 ab}{a + b}$ (Harmonic Mean)
Property: For distinct positive numbers, $A_{1} > G_{1} > H_{1}$ .

Defining the Iteration Rule

The iteration rule for $n \geq 2$ is:
$A_{n} = AM of A_{n - 1}, H_{n - 1} = \frac{A _{n - 1} + H _{n - 1}}{2}$
$G_{n} = GM of A_{n - 1}, H_{n - 1} = A_{n - 1} H_{n - 1}$
$H_{n} = HM of A_{n - 1}, H_{n - 1} = \frac{2 A _{n - 1} H _{n - 1}}{A _{n - 1} + H _{n - 1}}$

The Constancy of Geometric Mean

Calculate $G_{2}$ using the previous means:
$G_{2} = A_{1} H_{1}$
Recall the identity for any two numbers: $A M \times H M = G M^{2}$ .
Therefore, $A_{1} H_{1} = G_{1}^{2}$ .
Substituting this: $G_{2} = G_{1}^{2} = G_{1}$ .
By induction, $G_{n} = G_{n - 1} = \dots = G_{1}$ for all $n$ .

Analyzing the Arithmetic Mean Sequence

Examine the difference $A_{n} - A_{n - 1}$ :
$A_{n} - A_{n - 1} = \frac{A _{n - 1} + H _{n - 1}}{2} - A_{n - 1} = \frac{H _{n - 1} - A _{n - 1}}{2}$
Since $A_{n - 1} > H_{n - 1}$ for all $n$ , the term $(H_{n - 1} - A_{n - 1})$ is negative.
Thus, $A_{n} < A_{n - 1}$ .
This gives the decreasing sequence: $A_{1} > A_{2} > A_{3} > \dots$

Analyzing the Harmonic Mean Sequence

Relate $H_{n}$ to $A_{n}$ using the constant $G_{n}$ :
$H_{n} = \frac{G _{n}^{2}}{A _{n}} = \frac{G _{1}^{2}}{A _{n}}$
Since the sequence $A_{n}$ is decreasing ( $A_{1} > A_{2} > A_{3} > \dots$ ), the reciprocal sequence $\frac{1}{A _{n}}$ must be increasing.
Therefore, $H_{n}$ is an increasing sequence.
Conclusion: $H_{1} < H_{2} < H_{3} < \dots$

Summary and Convergence

Final Results:
1. $G_{1} = G_{2} = G_{3} = \dots$ (Constant)
2. $A_{1} > A_{2} > A_{3} > \dots$ (Decreasing)
3. $H_{1} < H_{2} < H_{3} < \dots$ (Increasing)
Key Takeaway: Both sequences $A_{n}$ and $H_{n}$ converge to the common limit $G_{1}$ as $n \to \infty$ .

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Conceptually Similar Problems

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Let $a_{1}, a_{2}, \dots a_{n}$ be positive real numbers in geometric progression. For each $n$ , let $A_{n}, G_{n}, H_{n}$ be respectively, the arithmetic mean, geometric mean, and harmonic mean of $a_{1}, a_{2}, \dots a_{n}$ . Find an expression for the geometric mean of $G_{1}, G_{2}, \dots G_{n}$ in terms of $A_{1}, A_{2}, \dots A_{n}, H_{1}, H_{2}, \dots H_{n}$ .

Answer:

G = (A_{1} A_{2} \dots A_{n} H_{1} H_{2} \dots H_{n})^{1/2 n}

MathematicsSum of Special SeriesJEE Main 2008Easy

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Statement-1 : For every natural number $n \geq 2, \frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n} > n$ . Statement-2 : For every natural number $n \geq 2, n (n + 1) < n + 1$ .

(A)

Statement -1 is false, Statement-2 is true

(B)

Statement -1 is true, Statement-2 is true; Statement -2 is a correct explanation for Statement-1

(C)

Statement -1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1

(D)

Statement -1 is true, Statement-2 is false

Answer:B

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Let $p$ be the first of the $n$ arithmetic means between two numbers and $q$ the first of $n$ harmonic means between the same numbers. Show that $q$ does not lie between $p$ and $(\frac{n + 1}{n - 1})^{2} p$ .

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Let the harmonic mean and geometric mean of two positive numbers be the ratio $4 : 5$ . Then the two number are in the ratio \dots.

Answer:

4 : 1 or 1 : 4

MathematicsGeometric Progression (G.P.)JEE Advanced 2008Easy

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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.

(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

Answer:C

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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:

(A)

4 l m n^{2}

(B)

4 l^{2} m^{2} n^{2}

(C)

4 l^{2} mn

(D)

4 l m^{2} n

Answer:D

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If $x_{1}, x_{2}, \dots, x_{n}$ are any real numbers and $n$ is any positive integer, then

(A)

n \sum_{i = 1}^{n} x_{i}^{2} < (\sum_{i = 1}^{n} x_{i})^{2}

(B)

\sum_{i = 1}^{n} x_{i}^{2} \geq (\sum_{i = 1}^{n} x_{i})^{2}

(C)

\sum_{i = 1}^{n} x_{i}^{2} \geq n (\sum_{i = 1}^{n} x_{i})^{2}

(D)

none of these

Answer:D

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Let $a, b$ be positive real numbers. If $a, A_{1}, A_{2}, b$ are in arithmetic progression, $a, G_{1}, G_{2}, b$ are in geometric progression and $a, H_{1}, H_{2}, b$ are in harmonic progression, show that $\frac{G _{1} G _{2}}{H _{1} H _{2}} = \frac{A _{1} + A _{2}}{H _{1} + H _{2}} = \frac{( 2 a + b ) ( a + 2 b )}{9 ab}$ .

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The harmonic mean of two numbers is $4$ . Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the two numbers.

Answer:

3 and 6 or 6 and 3

MathematicsRelation Between A.M., G.M., and H.M.JEE Advanced 2016Moderate

View in:English Hindi

Let $b_{i} > 1$ for $i = 1, 2, \dots, 101$ . Suppose $lo g_{e} b_{1}, lo g_{e} b_{2}, \dots, lo g_{e} b_{101}$ are in Arithmetic Progression (A.P.) with the common difference $lo g_{e} 2$ . Suppose $a_{1}, a_{2}, \dots, a_{101}$ are in A.P. such that $a_{1} = b_{1}$ and $a_{51} = b_{51}$ . If $t = b_{1} + b_{2} + \dots + b_{51}$ and $s = a_{1} + a_{2} + \dots + a_{51}$ , then

(A)

s > t

and

a_{101} > b_{101}

(B)

s > t

and

a_{101} < b_{101}

(C)

s < t

and

a_{101} > b_{101}

(D)

s < t

and

a_{101} < b_{101}

Answer:B

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?

Visualized Solution (English)

Visualizing the Initial Means

Defining the Iteration Rule

The Constancy of Geometric Mean

Analyzing the Arithmetic Mean Sequence

Analyzing the Harmonic Mean Sequence

Summary and Convergence

Conceptually Similar Problems

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Which one of the following statements is correct ?

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Let the harmonic mean and geometric mean of two positive numbers be the ratio $4 : 5$ . Then the two number are in the ratio \dots.

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