SATHEE: Sequences and Series 5 Question 9

Sequences and Series 5 Question 9

9. (i) The value of $x + y + z$ is 15 . If $a, x, y, z, b$ are in $A P$ while the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is $\frac{5}{3}$ . If $a, x, y, z, b$ are in HP, then find $a$ and $b$ .

(ii) If $x, y, z$ are in HP, then show that $\log (x + z) + \log (x + z - 2 y) = 2 \log (x - z)$ .

(1978, 3M)

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Answer:

Correct Answer: 9.(i) $a = 1, b = 9$

Solution:

(i) Now, $a + b = (a + x + y + z + b) - (x + y + z)$

$= \frac{5}{2} (a + b) - 15$

[since, $a, x, y, z$ are in AP]

$∴ Sum = \frac{5}{2} (a + b) \Rightarrow a + b = 10$

Since, $a, x, y, z, b$ are in HP, then $\frac{1}{a}, \frac{1}{x}, \frac{1}{y}, \frac{1}{z}, \frac{1}{b}$ are in AP.

Now, $\frac{1}{a} + \frac{1}{b} = \frac{1}{a} + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{b} - \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$

$= \frac{5}{2} \frac{1}{a} + \frac{1}{b} - \frac{5}{3}$

$\Rightarrow \frac{a + b}{a b} = \frac{10}{9} \Rightarrow a b = \frac{9 \times 10}{10}$ [from Eq. (i)]

$\Rightarrow a b = 9$

On solving Eqs. (i) and (ii), we get

$a = 1, b = 9$

(ii) $L H S = \log (x + z) + \log (x + z - 2 y)$

$\begin{aligned} = \log (x + z) + \log x + z - 2 \frac{2 x z}{x + z} ∵ y = \frac{2 x z}{x + z} \\ = \log (x + z) + \log \frac{(x - z)^{2}}{(x + z)} \\ = 2 \log (x - z) = RHS \end{aligned}$

Topic 6 Relation between AM, GM, HM and Some Special Series

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Sequences and Series 5 Question 9

9. (i) The value of $x + y + z$ is 15 . If $a, x, y, z, b$ are in $A P$ while the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is $\frac{5}{3}$ . If $a, x, y, z, b$ are in HP, then find $a$ and $b$ .

Answer:

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Sequences and Series 5 Question 9

9. (i) The value of x+y+z is 15 . If a,x,y,z,b are in AP while the value of 1x+1y+1z is 53. If a,x,y,z,b are in HP, then find a and b.

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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9. (i) The value of $x + y + z$ is 15 . If $a, x, y, z, b$ are in $A P$ while the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is $\frac{5}{3}$ . If $a, x, y, z, b$ are in HP, then find $a$ and $b$ .