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

The Correct Option is D

‘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.