JEE Advanced Maths – Harmonic Progression & AM GM HM Relation PYQ with Solutions

Q 1 :

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 [2012]

22
23
24
25

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(4)

$∵$ $a_{1}, a_{2}, a_{3}, \dots are in H.P.$

$∴$ $\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \dots are in A.P.$

$∴$ $\frac{1}{a_{1}} = \frac{1}{5} and \frac{1}{a_{20}} = \frac{1}{25}$

$\frac{1}{a_{1}} + 19 d = \frac{1}{a_{20}} \Rightarrow \frac{1}{5} + 19 d = \frac{1}{25} \Rightarrow d = \frac{- 4}{475}$

$Now \frac{1}{a_{n}} = \frac{1}{5} + (n - 1) (\frac{- 4}{475})$

$Clearly a_{n} < 0, if \frac{1}{a_{n}} < 0 \Rightarrow \frac{1}{5} - \frac{4 n}{475} + \frac{4}{475} < 0$

$\Rightarrow - 4 n < - 99$ or $n > \frac{99}{4} = 24 \frac{3}{4}$ $∴$ $n \geq 25$

$∴$ $Least value of n is 25 .$

Q 2 :

Let the positive numbers $a, b, c, d$ be in A.P. Then $a b c, a b d, a c d, b c d$ are [2001]

NOT in A.P./G.P./H.P.
in A.P.
in G.P.
in H.P.

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(4)

$a, b, c, d are in A.P.$ $∴$ $d, c, b, a are also in A.P.$

$\Rightarrow \frac{d}{a b c d}, \frac{c}{a b c d}, \frac{b}{a b c d}, \frac{a}{a b c d} are also in A.P.$

$\Rightarrow \frac{1}{a b c}, \frac{1}{a b d}, \frac{1}{a c d}, \frac{1}{b c d} are in A.P.$

$∴$ $a b c, a b d, a c d, b c d are in H.P.$

Q 3 :

Let $m$ be the minimum possible value of $\log_{3} (3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}}),$ where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1} + y_{2} + y_{3} = 9 .$ Let $M$ be the maximum possible value of $(\log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3}),$ where $x_{1}, x_{2}, x_{3}$ are positive real numbers for which $x_{1} + x_{2} + x_{3} = 9 .$ Then the value of $\log_{2} (m^{3}) + \log_{3} (M^{2})$ is _____ . [2020]

(8)

By AM--GM inequality

$AM \geq GM$

$∴$ $\frac{3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}}}{3} \geq {[3^{(y_{1} + y_{2} + y_{3})}]}^{\frac{1}{3}}$

$\Rightarrow 3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}} \geq 3^{4}$

$\Rightarrow \log_{3} (3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}}) \geq 4 \Rightarrow m = 4$

$∵$ $\log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3} = \log_{3} (x_{1} x_{2} x_{3})$

$Again by AM-GM inequality$

$AM \geq GM$

$\frac{x_{1} + x_{2} + x_{3}}{3} \geq \sqrt[3]{x_{1} x_{2} x_{3}} \Rightarrow x_{1} x_{2} x_{3} \leq 27$

$\Rightarrow \log_{3} (x_{1} x_{2} x_{3}) \leq \log_{3} (3^{3})$

$\Rightarrow \log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3} \leq 3 \Rightarrow M = 3$

$Now, \log_{2} (m^{3}) + \log_{3} (M^{2}) = 6 + 2 = 8$

Q 4 :

A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then [2008]

$\frac{1}{P S} + \frac{1}{S T} < \frac{2}{\sqrt{Q S \times S R}}$
$\frac{1}{P S} + \frac{1}{S T} > \frac{2}{\sqrt{Q S \times S R}}$
$\frac{1}{P S} + \frac{1}{S T} < \frac{4}{Q R}$
$\frac{1}{P S} + \frac{1}{S T} > \frac{4}{Q R}$

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Select one or more options

(2, 4)

$We know by geometry P S \times S T = Q S \times S R ...(i)$

$∵$ $S is not the centre of circumcircle, P S \neq S T$

And we know that for two unequal real numbers, H.M. < G.M.

$\Rightarrow \frac{2}{\frac{1}{P S} + \frac{1}{S T}} < \sqrt{P S \times S T} \Rightarrow \frac{1}{P S} + \frac{1}{S T} > \frac{2}{\sqrt{P S \times S T}}$

$\Rightarrow \frac{1}{P S} + \frac{1}{S T} > \frac{2}{\sqrt{Q S \times S R}} [using eqn (i)] ...(ii)$

$∴$ $(2) is the correct option.$

$Also \sqrt{Q S \times S R} < \frac{Q S + S R}{2} (∵ GM < AM)$

$\Rightarrow \frac{1}{\sqrt{Q S \times S R}} > \frac{2}{Q R} \Rightarrow \frac{2}{\sqrt{Q S \times S R}} > \frac{4}{Q R} ...(iii)$

$From equations (ii) and (iii), \frac{1}{P S} + \frac{1}{S T} > \frac{4}{Q R}$

$∴$ $(4) is also the correct option.$

Q 5 :

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. [2007]

Q. Which one of the following statements is correct?

$G_{1} > G_{2} > G_{3} > \dots$
$G_{1} < G_{2} < G_{3} < \dots$
$G_{1} = G_{2} = G_{3} = \dots$
$G_{1} < G_{3} < G_{5} < \dots$ and $G_{2} > G_{4} > G_{6} > \dots$

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(3)

$Given A_{1} = \frac{a + b}{2}, G_{1} = \sqrt{a b}, H_{1} = \frac{2 a b}{a + b}$

$Also A_{n} = \frac{A_{n - 1} + H_{n - 1}}{2}, G_{n} = \sqrt{A_{n - 1} H_{n - 1}}$

$H_{n} = \frac{2 A_{n - 1} H_{n - 1}}{A_{n - 1} + H_{n - 1}}$

$\Rightarrow G_{n}^{2} = A_{n} H_{n} \Rightarrow A_{n} H_{n} = A_{n - 1} H_{n - 1}$

$Similarly we can prove$

$A_{n} H_{n} = A_{n - 1} H_{n - 1} = A_{n - 2} H_{n - 2} = \dots = A_{1} H_{1}$

$\Rightarrow A_{n} H_{n} = a b$

$∴$ $G_{1}^{2} = G_{2}^{2} = G_{3}^{2} = \dots = a b$

$\Rightarrow G_{1} = G_{2} = G_{3} = \dots = \sqrt{a b}$

Q 6 :

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. [2007]

Q. Which one of the following statements is correct?

$A_{1} > A_{2} > A_{3} >$ ...
$A_{1} < A_{2} < A_{3} <$ ...
$A_{1} > A_{3} > A_{5} >$ ... and $A_{2} < A_{4} < A_{6} <$ ...
$A_{1} < A_{3} < A_{5} <$ ... and $A_{2} > A_{4} > A_{6} >$ ...

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(1)

$A_{n} = \frac{A_{n - 1} + H_{n - 1}}{2}$

$∴$ $A_{n} - A_{n - 1} = \frac{A_{n - 1} + H_{n - 1}}{2} - A_{n - 1}$

$= \frac{H_{n - 1} - A_{n - 1}}{2} < 0 (∵ A_{n - 1} > H_{n - 1})$

$\Rightarrow$ $A_{n} < A_{n - 1} or A_{n - 1} > A_{n}$

$∴$ $A_{1} > A_{2} > A_{3} >$ ...

Q 7 :

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. [2007]

Q. Which one of the following statements is correct?

$H_{1} > H_{2} > H_{3} >$ ...
$H_{1} < H_{2} < H_{3} <$ ...
$H_{1} > H_{3} > H_{5} >$ ... and $H_{2} < H_{4} < H_{6} <$ ...
$H_{1} < H_{3} < H_{5} <$ ... and $H_{2} > H_{4} > H_{6} >$ ...

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(2)

$A_{n} H_{n} = a b \Rightarrow H_{n} = \frac{a b}{A_{n}}$

$∵$ $\frac{1}{A_{n - 1}} < \frac{1}{A_{n}} \Rightarrow H_{n - 1} < H_{n}$

$∴$ $H_{1} < H_{2} < H_{3} <$ ....

Q 8 :

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.

STATEMENT-2: The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are in H.P. [2008]

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1
STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1
STATEMENT - 1 is True, STATEMENT - 2 is False
STATEMENT - 1 is False, STATEMENT - 2 is True

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(3)

$Given: a_{1}, a_{2}, a_{3}, a_{4} are in G.P.$

$Then b_{1}, b_{2}, b_{3}, b_{4} are the numbers a_{1}, a_{1} + a_{2}, a_{1} + a_{2} + a_{3}, a_{1} + a_{2} + a_{3} + a_{4}$

$or a, a + a r, a + a r + a r^{2}, a + a r + a r^{2} + a r^{3}$

Since, above numbers are neither in A.P. nor in G.P. Therefore,

statement 1 is true.

$Also \frac{1}{a}, \frac{1}{a + a r}, \frac{1}{a + a r + a r^{2}}, \frac{1}{a + a r + a r^{2} + a r^{3}} are not in A.P.$

$∴$ $b_{1}, b_{2}, b_{3}, b_{4} are not in H.P.$

$∴$ Statement 2 is false.

Topic Question Set

Q 1 :

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 [2012]

Q 2 :

Let the positive numbers $a, b, c, d$ be in A.P. Then $a b c, a b d, a c d, b c d$ are [2001]

Q 4 :

A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then [2008]

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Topic Question Set

Q 1 : Let a1,a2,a3,… be in harmonic progression with a1=5 and a20=25. The least positive integer n for which an<0 is [2012]

Q 2 : Let the positive numbers a,b,c,d be in A.P. Then abc, abd, acd, bcd are [2001]

Q 3 : Let m be the minimum possible value of log3(3y1+3y2+3y3), where y1,y2,y3 are real numbers for which y1+y2+y3=9. Let M be the maximum possible value of (log3x1+log3x2+log3x3), where x1,x2,x3 are positive real numbers for which x1+x2+x3=9. Then the value of log2(m3)+log3(M2) is _____ . [2020]

Q 4 : A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then [2008]

Q 5 : Let A1,G1,H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n≥2, let An-1 and Hn-1 have arithmetic, geometric and harmonic means as An,Gn,Hn respectively. [2007] Q. Which one of the following statements is correct?

Q 6 : Let A1,G1,H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n≥2, let An-1 and Hn-1 have arithmetic, geometric and harmonic means as An,Gn,Hn respectively. [2007] Q. Which one of the following statements is correct?

Q 7 : Let A1,G1,H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n≥2, let An-1 and Hn-1 have arithmetic, geometric and harmonic means as An,Gn,Hn respectively. [2007] Q. Which one of the following statements is correct?

Q 8 : Suppose four distinct positive numbers a1,a2,a3,a4 are in G.P. Let b1=a1, b2=b1+a2, b3=b2+a3, and b4=b3+a4. STATEMENT-1: The numbers b1,b2,b3,b4 are neither in A.P. nor in G.P. STATEMENT-2: The numbers b1,b2,b3,b4 are in H.P. [2008]

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Q 1 :

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 [2012]

Q 2 :

Let the positive numbers $a, b, c, d$ be in A.P. Then $a b c, a b d, a c d, b c d$ are [2001]

Q 4 :

A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then [2008]