JEE Main 2023 (Online) 11th April Morning ShiftQuestion 1
If $a$ and $b$ are the roots of the equation $x^{2}-7x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to $\_\_\_\_\_\_$ .
Quadratic Equation · Powers of Roots (αⁿ ± βⁿ Recurrence)
Answer$51$
Solution
Concept. $a^2=7a+1$ gives the recurrence $L_{k+2}=7L_{k+1}+L_k$ for $L_k=a^k+b^k$; use it to write the numerator through $L_{19}$. Formulas used. (1) $L_{k+2}=7L_{k+1}+L_k$Step 1 - $L_{21}$ and $L_{17}$. By (1), $L_{21}=7L_{20}+L_{19}$ and $L_{19}=7L_{18}+L_{17}$, so $$L_{21}+L_{17}=7(L_{20}-L_{18})+2L_{19}.$$ Step 2 - $L_{20}-L_{18}$. By (1), $L_{20}=7L_{19}+L_{18}$, so $L_{20}-L_{18}=7L_{19}$. Step 3 - Value. The numerator is $49L_{19}+2L_{19}=51L_{19}$, so the ratio is $51$. Answer: 51