Question

設(shè)z=1+i+i²+i³+…+i²⁰¹⁰，則

.

z

=______．

Answer 1

∵

in

in-1

=i，
∴{i^n-1}為首項(xiàng)為1，公比為i的等比數(shù)列，又i⁴ⁿ=1，i³=-i
∴z=1+i+i²+i³+…+i²⁰¹⁰=

1-i2011

1-i

=

1-i3

1-i

=

1+i

1-i

=

(1+i)2

(1-i)(1+i)

=i，
∴

.

z

=-i．