The maximum value of n<101 such that 1+k=1∑nik=0 is

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The maximum value of n < 101 such that 1 + _{k = 1}^{n}i^{k}

Question

The maximum value of $n < 101$ such that $1 + k = 1 \sum n i^{k} = 0$ is

96

97

99

100

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Eleanor Discussed

The maximum value of

n < 101

such that

1 + k = 1 \sum n i^{k} = 0

6 mins ago

Discuss this question LIVE

6 mins ago

\sum_{k = 1}^{n} i^{k} = i + i^{2} + i^{3} + i^{4} + i^{5} + i^{6} + i^{7} + i^{8} + i^{9} + ....

Now, as

i^{4} = i^{8} = i^{12} = ... i^{4 n} = 1

And

i + i^{2} + i^{3} + i^{4} = 0

So,

\sum_{k = 1}^{n} i^{k} = i + i^{2} + i^{3} + i^{4} + i^{5} + i^{6} + i^{7} + i^{8} + i^{9} + ....

= i + i^{2} + i^{3} + i^{4} + (i + i^{2} + i^{3} + i^{4}) + (i + i^{2} + i^{3} + i^{4}) + ...

= 0 + 0 + 0....

When we take

n = 99

after

96

th term, we will be left with

0 + i + i^{2} + i^{3} = - 1

So, for

n = 99

1 + k = 1 \sum n i^{k} = 0

Hence, option C is correct.

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Question Text	The maximum value of $n < 101$ such that $1 + k = 1 \sum n i^{k} = 0$ is
Answer Type	Text solution:1
Upvotes	150