Question

The sequence $S=i+2i^2+3i^3+......upto100$ terms simplifies to where $i=\sqrt{-1}$

• A. $50(1-i)$
• B. $25i$
• C. $25(1+i)$
• D. $100(1-i)$

Answer 1

The correct option is A $50(1-i)$

$S=i+2i^2+3i^3+...100i^{100}$......(i)

$Si=i^2+2i^3+...99i^{100}+100i^{101}$.....(ii)

$Eq\left(i\right)-Eq\left(ii\right)$

$S(1-i)=i+i^2+i^3+...i^{100}-100i^{101}$

$S_n=\frac{a(1-r^n)}{1-r}$........Sum of n terms (GP)

$a=i,r=i,n=100$

$S(1-i)=\frac{i(1-i^{100})}{1-i}-100\left(i^{100}\right).i$

$S(1-i)=0-100i$

$S=100\left(\frac{-i}{1-i}\right)$

$=100\left(\frac{-i(1+i)}{2}\right)$

$=50(1-i)$.