SATHEE: Sequences and Series 2 Question 7

Sequences and Series 2 Question 7

8.

Let $b_{i} > 1$ for $i = 1, 2, \dots, 101$ . Suppose $\log_{e} b_{1}, \log_{e} b_{2}$ , $\dots, \log_{e} b_{101}$ are in AP with the common difference $\log_{e} 2$ . Suppose $a_{1}, a_{2}, \dots, a_{101}$ are in AP, 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

(2016 Adv.)

(a) $s > t$ and $a_{101} > b_{101}$

(b) $s > t$ and $a_{101} < b_{101}$

(d) $s < t$ and $a_{101} < b_{101}$

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Answer:

Correct Answer: 8. (b)

Solution:

If $\log b_{1}, \log b_{2}, \dots, \log b_{101}$ are in $A P$ , with common difference $\log_{e} 2$ , then $b_{1}, b_{2}, \dots, b_{101}$ are in GP, with common ratio 2 .

$∴ b_{1} = 2^{0} b_{1}, b_{2} = 2^{1} b_{1}, b_{3} = 2^{2} b_{1}, \dots, b_{101} = 2^{100} b_{1}$

Also, $a_{1}, a_{2}, \dots, a_{101}$ are in AP.

Given, $a_{1} = b_{1}$ and $a_{51} = b_{51}$

$\begin{array}{llll} \Rightarrow & a_{1} + 50 D = 2^{50} b_{1} \\ \Rightarrow & a_{1} + 50 D = 2^{50} a_{1} [∵ a_{1} = b_{1}] . \end{array}$

Now, $t = b_{1} + b_{2} + \dots + b_{51}$

$\Rightarrow t = b_{1} \frac{(2^{51} - 1)}{2 - 1}$

and $s = a_{1} + a_{2} + \dots + a_{51}$

$= \frac{51}{2} (2 a_{1} + 50 D)$

$∴ t = a_{1} (2^{51} - 1) [∵ a_{1} = b_{1}]$

or $t = 2^{51} a_{1} - a_{1} < 2^{51} a_{1}$

and $s = \frac{51}{2} [a_{1} + (a_{1} + 50 D)]$ [from Eq. (ii)]

$= \frac{51}{2} [a_{1} + 2^{50} a_{1}]$

$= \frac{51}{2} a_{1} + \frac{51}{2} 2^{50} a_{1}$

$∴ s > 2^{51} a_{1}$

From Eqs. (v) and (vi), we get $s > t$

Also, $a_{101} = a_{1} + 100 D$ and $b_{101} = 2^{100} b_{1}$

$\begin{aligned} ∴ a_{101} = a_{1} + 100 \frac{2^{50} a_{1} - a_{1}}{50} and b_{101} = 2^{100} a_{1} \\ \Rightarrow a_{101} = a_{1} + 2^{51} a_{1} - 2 a_{1} = 2^{51} a_{1} - a_{1} \\ \Rightarrow a_{101} < 2^{51} a_{1} and b_{101} > 2^{51} a_{1} \\ \Rightarrow b_{101} > a_{101} \end{aligned}$

Sequences and Series 2 Question 7

8.

Answer:

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Sequences and Series 2 Question 7

8.

Answer:

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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