Question

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

Short answer

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

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".

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.

Finding the answer

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

Key concepts

Imaginary Unit

Understanding the concept of the imaginary unit is fundamental when dealing with complex numbers. The imaginary unit is denoted by the symbol \( i \). It is defined by the property that its square is equal to -1, meaning \( i^2 = -1 \).
This concept is crucial because it extends our number system to include complex numbers, which are combinations of real and imaginary numbers.
Here is what you need to remember about the imaginary unit:

• Imaginary numbers can be thought of as multiples of \( i \).
• The imaginary unit enables us to handle roots of negative numbers, which are not solvable using only real numbers. For example, the square root of -1 is \( i \).
• Every imaginary unit multiplied by any real number still gives us an imaginary number, like 3\( i \) or -5\( i \).

Imaginary units simplify the algebra involving complex numbers and play a critical role in many mathematical theories and applications.

Powers of i

When working with powers of the imaginary unit \( i \), a fascinating pattern emerges. This pattern repeats every four powers of \( i \), aiding in simplifying many calculations. Let's break down this cycle:

• \( i^1 = i \)
• \( i^2 = -1 \)
• \( i^3 = -i \)
• \( i^4 = 1 \)

After completing one cycle, the powers repeat. Thus:

• \( i^5 = i \)
• \( i^6 = -1 \)
• \( i^7 = -i \)
• \( i^8 = 1 \)

The recurring cycle of powers of \( i \) is essential for solving complex number problems.
For example, in the exercise, identifying this cycle helps in determining that the entire sum of \( i^1 + i^2 + ... + i^{1000} \) equals zero because each full cycle of 4 terms sums to zero.

Sequence Patterns

The discovery and application of sequence patterns are particularly beneficial in dealing with complex numbers like the powers of \( i \). Identifying patterns can make apparently large computations much simpler.
To illustrate this, consider the sequence of powers of \( i \) discussed earlier. Recognizing that every four powers of \( i \) repeat allows us to simplify calculations into manageable parts.
For example:

• In the given problem, we divided 1000 by 4, because each cycle contains 4 terms. This resulted in 250 complete cycles.
• Knowing that each cycle sums to zero (\( i - 1 - i + 1 = 0 \)), it becomes straightforward to deduce that the entire sum for \( i^{1000} \) is zero as well.

This method of identifying repeating cycles and patterns is widely used in different fields of mathematics and is extremely effective in reducing complexity in calculations.