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Chapter 2: Problem 103

\(\mathrm{i}^{1}+\mathrm{i}^{2}+\mathrm{i}^{3}+\mathrm{i}^{4}+\ldots \ldots \ldots \mathrm{i}^{1000}=\) (a) \(-1\) (b) 0 (c) 1 (d) None

Short Answer

Expert verified
The sum of the series \(\mathrm{i}^{1}+\mathrm{i}^{2}+\mathrm{i}^{3}+\mathrm{i}^{4}+\cdots + \mathrm{i}^{1000}\) is 0, as there are 250 complete cycles in the sum with each cycle having a sum of 0. The correct answer is (b) 0.

Step by step solution

01

Identifying the pattern of powers of i

To find the pattern, we'll list down the powers of i and see if there's a repeating cycle: \(\mathrm{i}^{1} = i\) \(\mathrm{i}^{2} = -1\) \(\mathrm{i}^{3} = i(-1) = -i\) \(\mathrm{i}^{4} = (-1)^{2} = 1\) \(\mathrm{i}^{5} = \mathrm{i}\cdot \mathrm{i}^{4} = i\) \(\mathrm{i}^{6} = \mathrm{i}^{2}\cdot \mathrm{i}^{4} = -1\) We can observe that there is a repeating cycle in the powers of i every 4 powers, i.e., the pattern is "i, -1, -i, 1".
02

Calculating the sum of the series with the pattern

As we know that the sum has 1000 terms, let's divide 1000 by the length of the pattern (4) to see the number of complete cycles in the sum: \(1000 / 4 = 250\) This tells us that there are 250 complete cycles in the sum of the series. Since each complete cycle has a sum of 0, i.e., (i - 1 - i + 1) = 0, this implies that the sum of all the 1000 terms is also 0.
03

Finding the answer

As the sum of the series is 0, the correct answer is: (b) 0

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