If a and b are the roots of the equation x 2 - 7 x - 1 = 0 , then the value of a 21 + b 21 + a 17 + b 17 a 19 + b 19 is equal to ____ . [2023]

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Solution

Q.
If $a$ and $b$ are the roots of the equation $x^{2} - 7 x - 1 = 0$ , then the value of $\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}}$ is equal to ____ . [2023]

Ans.
(51)

$Given: a and b are the roots of x^{2} - 7 x - 1 = 0$

$By Newton's theorem,$

$S_{n + 2} - 7 S_{n + 1} - S_{n} = 0$

$S_{21} - 7 S_{20} - S_{19} = 0 ...(i)$

$S_{20} - 7 S_{19} - S_{18} = 0 ...(ii)$

$S_{19} - 7 S_{18} - S_{17} = 0 ...(iii)$

$\frac{S_{21} + S_{17}}{S_{19}} = \frac{S_{21} + (S_{19} - 7 S_{18})}{S_{19}} [From (iii)]$

$= \frac{S_{21} + S_{19} - 7 (S_{20} - 7 S_{19})}{S_{19}} [From (ii)]$

$= \frac{50 S_{19} + (S_{21} - 7 S_{20})}{S_{19}} = \frac{50 S_{19} + S_{19}}{S_{19}} [From (i)]$

$= 51 \cdot \frac{S_{19}}{S_{19}} = 51$

$∴$ $\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}} = 51$

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