SATHEE: Sequences and Series 2 Question 23

Sequences and Series 2 Question 23

24.

Let $a_{1}, a_{2}, a_{3}, \dots, a_{11}$ be real numbers satisfying $a_{1} = 15$ , $27 - 2 a_{2} > 0$ and $a_{k} = 2 a_{k - 1} - a_{k - 2}$ for $k = 3, 4, \dots, 11$ . If $\frac{a_{1}^{2} + a_{2}^{2} + \dots + a_{11}^{2}}{11} = 90$ , then the value of

$\frac{a_{1} + a_{2} + \dots + a_{11}}{11}$ is……

(2010)

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Answer:

Correct Answer: 24. $(0)$

Solution:

$a_{k} = 2 a_{k - 1} - a_{k - 2}$

$\Rightarrow a_{1}, a_{2}, \dots, a_{11}$ are in an AP.

$∴ \frac{a_{1}^{2} + a_{2}^{2} + \dots + a_{11}^{2}}{11} = \frac{11 a^{2} + 35 \times 11 d^{2} + 10 a d}{11} = 90$

$\Rightarrow 225 + 35 d^{2} + 150 d = 90$

$\Rightarrow 35 d^{2} + 150 d + 135 = 0 \Rightarrow d = - 3, - \frac{9}{7}$

Given, $a_{2} < \frac{27}{2}$

$∴ d = - 3$ and $d \neq - \frac{9}{7}$

$\Rightarrow \frac{a_{1} + a_{2} + \dots + a_{11}}{11} = \frac{11}{2} [30 - 10 \times 3] = 0$

Sequences and Series 2 Question 23

24.

Answer:

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Sequences and Series 2 Question 23

24.

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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