Question

If $\sum _{k=0}^{200}{i}^k+{\Pi }_{p=1}^{50}{i}^p=x+iy$ then
the values of x and y are x $=$ ................. and y $=$ ................

• A. $x=1,y=1$
• B. $x=1,y=-1$
• C. $x=-1,y=1$
• D. $x=-1,y=-1$

Answer 1

The correct option is B $x=1,y=-1$

${\sum }_{k=0}^{200}{i}^k+{\Pi }_{p=1}^{50}{i}^p=x+iy$
$\Rightarrow {\sum }_{k=0}^{200}{i}^k+i^{{\sum }_{p=1}^{50}p}=x+iy$
$\Rightarrow \frac{i^{201}-1}{i-1}+i^{\frac{50(50+1)}{2}}=\frac{{\left(i^4\right)}^{50}i-1}{i-1}+i^{1275}=1+{\left(i^2\right)}^{637}i=1-i=x+iy$
Therefore, $x=1,y=-1$
Ans: B