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4-1.Complex numbers
easy
If $\sum\limits_{k = 0}^{100} {{i^k}} = x + iy$, then the values of $x$ and $y$are
A
$x = - 1,y = 0$
B
$x = 1,y = 1$
C
$x = 1,y = 0$
D
$x = 0,y = 1$
Solution
(c) $\sum\limits_{k = 0}^{100} {{i^k} = x + iy,} $==> $1 + i + {i^2}$$ + …… + {i^{100}} = x + iy$
Given series is G.P.
==> $\frac{{1.(1 – {i^{101}})}}{{1 – i}} = x + iy$ ==> $\frac{{1 – i}}{{1 – i}} = x + iy$
==> $1 + 0i = x + iy$
Equating real and imaginary parts, we get the required result.
Standard 11
Mathematics