Question

If $x^2+x+1=0$, then value of
${(x+\frac{1}{x})}^3+{(x^2+\frac{1}{x^2})}^3+...+{(x^{100}+\frac{1}{x^{100}})}^3$
is

• A. $100$
• B. $197$
• C. $50$
• D. $97$

Answer 1

The correct option is B $197$

The roots of the equation are $\omega$ and ${\omega }^2$
We substitute $x$ with any one of the two roots and try to find the answer.
We have $x+\frac{1}{x}=\omega +{\omega }^2=-1$
Thus, all the terms which don't have powers in multiple of $3$, will give the value $-1$ and $(-1{)}^3=-1$.
On the other hand, the terms with power of $x$ being a multiple of $3$ will give $2$ and $(2{)}^3=8$
Thus, we have in all 33 terms with power divisible by 3 and thus the answer looks like $33\times 8+67\times (-1)=264-67=197$