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^…

Question

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 \_\_\_\_\_\_\_\_\_\_\_\_\_.

Answer: 51

Accepted Answer

We have, $a$ and $b$ are the roots of the equation

$\begin{gathered} {x}^{2}-7x-1=0 \ \Rightarrow {a}^{2}-7a-1=0\Rightarrow {a}^{2}-1=7a.........(i) \end{gathered}$

On squaring both sides, we get ${a}^{4}+1=51{a}^{2}$

Similarly, ${b}^{4}+1=51{b}^{2}$ ...........(ii)

$\frac{{a}^{21}+{b}^{21}+{a}^{17}+{b}^{17}}{{a}^{19}+{b}^{19}}=\frac{{a}^{17}({a}^{4}+1)+{b}^{17}({b}^{4}+1)}{{a}^{19}+{b}^{19}}$

$\begin{gathered} =\frac{{a}^{17}(51{a}^{2})+{b}^{17}(51{b}^{2})}{{a}^{19}+{b}^{19}}\, [\because ] \ =\frac{51[{a}^{19}+{b}^{19}]}{{a}^{19}+{b}^{19}}=51 \end{gathered}$

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