Question

find the value of -
1+ i +i^2 + i^3 + ..... + i^101​

Answer 1

Answer:i

Step-by-step explanation:

Answer is in photo below hope it helps

Answer 2

The correct answer is i+1.

Given: $1+i+i^{2}............+i^{101}$.

To Find: The summation of this series.

Solution:

$1+i+i^{2}............+i^{101}$  is a GP with i as common ratio and first term is 1.

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

1 can be written as $i^{0}$.

So this term has 102 terms.

Sum of GP = $\frac{1(i^{102}-1 )}{i-1}$

= $i^{102}=i^{100+2}=i^{2} =-1$

Sum of GP = $\frac{-1-1}{i-1}$

= $\frac{-2}{i-1}$

Rationalize the term.

= $\frac{-2}{i-1}*\frac{i+1}{i+1}$

= $\frac{-2i-2}{i^{2} -1}$

= $\frac{-2i-2}{-1 -1}$

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

= i+1

Hence, the answer is i+1.

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