SATHEE: Theory of Equations 1 Question 56

Theory of Equations 1 Question 56

Passage Type Questions

Let $p, q$ be integers and let $α, β$ be the roots of the equation, $x^{2} - x - 1 = 0$ where $α \neq β$ . For $n = 0, 1, 2, \dots \dots$ , let $a_{n} = p α^{n} + q β^{n}$ .

FACT : If $a$ and $b$ are rational numbers and $a + b \sqrt{5} = 0$ , then $a = 0 = b$ .

(2017 Adv.)

57. $a_{12} =$

(a) $a_{11} + 2 a_{10}$

(b) $2 a_{11} + a_{10}$

(d) $a_{11} + a_{10}$

Show Answer

Answer:

Correct Answer: 57. (d)

Solution:

$α^{2} = α + 1$

$β^{2} = β + 1$

$a_{n} = p α^{n} + q β^{n}$

$= p (α^{n - 1} + α^{n - 2}) + q (β^{n - 1} + β^{n - 2})$

$= a_{n - 1} + a_{n - 2}$

$∴ a_{12} = a_{11} + a_{10}$

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Theory of Equations 1 Question 56

Passage Type Questions

57. a12=

Answer:

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57. $a_{12} =$