Question

Let a and b are roots of x2 – 7x – 1 = 0. The value of $\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}}$ is?

• A. 29
• B. 49
• C. 53
• D. 51

Answer 1

Answer: (D)

51

Answer 2

‘a’ and ‘b’ are roots of $x^2 -7x -2 =0$ to find $\frac{a^{17}( a^4+1) + b^{17}(b^4 + 1) }{a^{19} + b^{19}}$
Considering one of the root ‘$\alpha$’ for the equation;
$\alpha ^2 - 1 = 7\alpha$
⇒ $\alpha ^4 + 1 = 51\alpha ^2$
∴$\frac{51a^{19} + 51b^{29}}{a^{19}+ b^{19}}$ [Here, consider as \large\alpha^2$$\large<^{\large{a}}_{\large{b}} ]
$=51(\frac{a^{19} + b^{29}}{a^{19}+ b^{19}})$
$=51$
Hence, The correct answer is the option (D) 51.