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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
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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
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