Prove that (1 + i² + i⁴ + i⁶ + i⁸ + …. + i²⁰) = 1.

L.H.S = (1 + i² + i⁴ + i⁶ + i⁸ + …. + i²⁰)

=

= 1 + -1 +1 + -1 + ……….. + 1

As there are 11 times 1 and 6 times it is with positive sign as =1 as this is the extra term and there are 5 times 1 with negative sign

So, these 5 cancel out the positive one leaving one positive value i.e. 1

=

L.H.S = R.H.S

Hence proved.