If a and b are the roots of the equation x²-7 x-1=0, then the value of (a²¹+b²¹+a¹⁷+b¹⁷)/(a¹⁹+b¹⁹) is equal to _____________.

Solution

Maths sir · 4 steps

Do not expand the powers: pull out a¹⁹, and ab = -1 turns the leftover into a² + b².

1. Read the roots. For x² - 7x - 1 = 0, the sum of roots is a + b = 7 and the product is ab = -1. So 1/a = -b and 1/b = -a, which gives a⁻² = b² and b⁻² = a².
2. Factor each root. Pull a¹⁹ out of the a terms: a²¹ + a¹⁷ = a¹⁹(a² + a⁻²) = a¹⁹(a² + b²). In the same way, b²¹ + b¹⁷ = b¹⁹(b² + b⁻²) = b¹⁹(a² + b²).

   (a²¹+b²¹+a¹⁷+b¹⁷)/(a¹⁹+b¹⁹) = (a²+b²)(a¹⁹+b¹⁹)/(a¹⁹+b¹⁹) = a² + b²

3. Find the sum. a² + b² = (a+b)² - 2ab = 49 - 2(-1) = 49 + 2 = 51.
4. So. The whole expression equals a² + b², which is 51.

The common slip: 47. Taking the product of roots as +1 instead of -1 gives a² + b² = 49 - 2 = 47, not 51.

The idea to keep. When the powers in a ratio are spaced evenly, factor out the middle power of each root. If ab = ± 1, the reciprocal of one root is the other root up to a sign.

Answer from the NTA final answer key. Solution by PrepWiser.