SATHEE: Sequences and Series 5 Question 17

Sequences and Series 5 Question 17

8. Let $a$ and $b$ 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, then show that

$\frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}} = \frac{(2 a + b) (a + 2 b)}{9 a b}$

(2002, 5M)

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Solution:

Since, $a, A_{1}, A_{2}, b$ are in AP.

$\begin{array}{ll} \Rightarrow & A_{1} + A_{2} = a + b \\ a, G_{1}, G_{2}, b are in GP \Rightarrow G_{1} G_{2} = a b \end{array}$

and $a, H_{1}, H_{2}, b$ are in HP.

$\begin{array}{ll} \Rightarrow & H_{1} = \frac{3 a b}{2 b + a}, H_{2} = \frac{3 a b}{b + 2 a} \\ ∴ & \frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{1}{a} + \frac{1}{b} \\ \Rightarrow & \frac{H_{1} + H_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{G_{1} G_{2}} = \frac{1}{a} + \frac{1}{b} \end{array}$

Now,

$\begin{aligned} \frac{G_{1} G_{2}}{H_{1} H_{2}} & = \frac{a b}{\frac{3 a b}{2 b + a} \frac{3 a b}{b + 2 a}} \\ = \frac{(2 a + b) (a + 2 b)}{9 a b} \end{aligned}$

From Eqs. (i) and (ii), we get

$\frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}} = \frac{(2 a + b) (a + 2 b)}{9 a b}$

Sequences and Series 5 Question 17

8. Let $a$ and $b$ 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, then show that

Solution:

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Sequences and Series 5 Question 17

8. Let a and b be positive real numbers. If a,A1,A2,b are in arithmetic progression, a,G1,G2,b are in geometric progression and a,H1,H2,B are in harmonic progression, then show that

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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8. Let $a$ and $b$ 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, then show that