If t2+t+1=0 , prove (t+1/t)2+(t2+1/t2)2+..+(t27+1/t27)2=54, Mathematics

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t=w,w²

L.H.S

(w+w²) + (w²+ w)² ........

1 + 1 .....

But every third term is of the form:

(w³ⁿ +w³ⁿ)²=22

There are nine such terms. Their sum is 36.

The remaining are one. Their sum is 18.

Adding them gives 54.

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