If x2 1 x2 1 then what is the value of x48 x42 x36 x30 x24 x18 x1...........

Quantitative Aptitude (CGL) Algebra

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If x² + (1/x²) = 1, then what is the value of x⁴⁸ + x⁴² + x³⁶ + x³⁰ + x²⁴ + x¹⁸ + x¹² + x⁶ + 1?

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Explanation:

Using the fact:
if x+1/x=√3
then,
x^6 = -1
If x^2+1/x^2=1
First find: x+1/x
(x+1/x)^2=x^2+1/x^2+2
Then;
(x+1/x)^2=1+2=3
=>x+1/x=√x
⇒x^6 = -1
We have to find the value of x^{48} + x^{42} + x^{36}+ x^{30}+ x^{24} + x^{18} + x^{12} + x^{6}+1:
We can write this as:
(x^{6})^8+ (x^{6})^7 + (x^{6})^6+ (x^{6})^5+ (x^{6})^4 + (x^{6})^3+ (x^{6})^2 + x^{6}+1:
Substitute x^6 = -1 then;
(-1)^8+(-1)^7+(-1)^6+(-1)^5+(-1)^4+(-1)^3+(-1)^2+(-1)+1
⇒1-1+1-1+1-1+1-1+1
Simplify:
⇒1