Question

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Answer 1

Let alpha, beta be the roots of x^2 - x - 1 = 0, if an = a^n + b^n (n ≥ 1), then the value of (a12 - a11)/a10 is

Answer:

1

Step-by-step explanation:

Here, if a & b are roots.

Sum of roots = a + b = 1

a = b - 1 & b = 1 - a ... (1)

Product of roots = ab = - 1 ... (2)

Now,

a(12) - a(11) = a¹² + b¹² - (a¹¹ + b¹¹)

= a¹² - a¹¹ + b¹² - b¹¹

= a¹¹(a - 1) + b¹¹(b - 1)

= a¹¹{-(1 - a)} + b¹¹{-(1 - b)}

= - {a¹¹(1 - a) + b¹¹(1 - b)}

From (1), = - {a¹¹b + b¹¹a}

= - ab(a^10 + b^10)

Hence,

=> (a12 - a11)/(a10)

=> - ab(a^10 + b^10)/(a^10 + b^10)

=> - ab from(2)

=> -(-1) => 1

Note that ^10 refers to power. a^10 refers to a to the power 10, in the same manner as a^2 refers to a². Wherever possible I have used power directly.