x²+x+1=0 LHS is sum of a geometric series with a=1 and r=x. The nth element is x^(n-1). Note that for any n the sum of any three consecutive elements: x^(n-1)+x^n+x^(n+1) =[x^(n-1)](1+x+x²)=0 Thus sum of any three consecutive elemennts is 0. Let a=1+x+x²+...+x⁴⁸+x⁴⁹+x⁵⁰+...+x⁵³ There are 54 elements, therefore a=0 as 3|54. Let b is sum up to x⁴⁸. There are 49 elements. Taking 3 consecutive elements from x⁴⁸, it leaves 1, the first element. Thus b=1 x⁴⁹+x⁵⁰+x⁵¹+x⁵²+x⁵³=a-b =0-1=-1
Why do you start by stating that x is real? X must be complex.
Nice example
This can be easily done using cube roots of unity.
x²+x+1=0
LHS is sum of a geometric series with a=1 and r=x. The nth element is x^(n-1).
Note that for any n the sum of any three consecutive elements:
x^(n-1)+x^n+x^(n+1)
=[x^(n-1)](1+x+x²)=0
Thus sum of any three consecutive elemennts is 0.
Let
a=1+x+x²+...+x⁴⁸+x⁴⁹+x⁵⁰+...+x⁵³
There are 54 elements, therefore a=0 as 3|54.
Let b is sum up to x⁴⁸. There are 49 elements. Taking 3 consecutive elements from x⁴⁸, it leaves 1, the first element. Thus b=1
x⁴⁹+x⁵⁰+x⁵¹+x⁵²+x⁵³=a-b
=0-1=-1
Shouldn't X be complexe ?
It will be more beautiful !💛