The value of i 2n + i 2n+1 + i 2n+2 + i 2n+3 , where \({\rm{i}} = \sqrt { - 1} \) , is
0
Understanding the Problem: Value of Consecutive Powers of i
The question asks us to find the value of the expression \({\rm{i}}^{2n} + {\rm{i}}^{2n+1} + {\rm{i}}^{2n+2} + {\rm{i}}^{2n+3}\), where \({\rm{i}} = \sqrt{-1}\). This expression involves the imaginary unit \({\rm{i}}\) raised to consecutive powers.
The imaginary unit \({\rm{i}}\) is a fundamental concept in complex numbers. Its powers follow a specific repeating pattern.
Key Property of Powers of i
The powers of the imaginary unit \({\rm{i}}\) cycle through four values:
- \({\rm{i}}^1 = {\rm{i}}\)
- \({\rm{i}}^2 = -1\)
- \({\rm{i}}^3 = {\rm{i}}^2 \times {\rm{i}} = -1 \times {\rm{i}} = -{\rm{i}}\)
- \({\rm{i}}^4 = ({\rm{i}}^2)^2 = (-1)^2 = 1\)
This cycle \({\rm{i}}, -1, -{\rm{i}}, 1\) repeats for higher powers. For example, \({\rm{i}}^5 = {\rm{i}}^4 \times {\rm{i}} = 1 \times {\rm{i}} = {\rm{i}}\), \({\rm{i}}^6 = -1\), \({\rm{i}}^7 = -{\rm{i}}\), \({\rm{i}}^8 = 1\), and so on.
Solving the Expression with Powers of i
The given expression is the sum of four consecutive powers of \({\rm{i}}\): \({\rm{i}}^{2n}, {\rm{i}}^{2n+1}, {\rm{i}}^{2n+2}, {\rm{i}}^{2n+3}\).
We can factor out the term with the lowest exponent, \({\rm{i}}^{2n}\), from the entire expression:
\[{\rm{i}}^{2n} + {\rm{i}}^{2n+1} + {\rm{i}}^{2n+2} + {\rm{i}}^{2n+3} = {\rm{i}}^{2n}(1) + {\rm{i}}^{2n}({\rm{i}}^{(2n+1) - 2n}) + {\rm{i}}^{2n}({\rm{i}}^{(2n+2) - 2n}) + {\rm{i}}^{2n}({\rm{i}}^{(2n+3) - 2n})\]
\[= {\rm{i}}^{2n}(1 + {\rm{i}}^1 + {\rm{i}}^2 + {\rm{i}}^3)\]
Now, let's evaluate the sum of the powers inside the parenthesis: \(1 + {\rm{i}}^1 + {\rm{i}}^2 + {\rm{i}}^3\).
Substitute the known values for these powers of \({\rm{i}}\):
\[1 + {\rm{i}}^1 + {\rm{i}}^2 + {\rm{i}}^3 = 1 + ({\rm{i}}) + (-1) + (-{\rm{i}})\]
\[= 1 + {\rm{i}} - 1 - {\rm{i}}\]
Group the real and imaginary terms:
\[= (1 - 1) + ({\rm{i}} - {\rm{i}})\]
\[= 0 + 0\]
\[= 0\]
The sum of the terms inside the parenthesis is \(0\).
Now, multiply this result by the factored term, \({\rm{i}}^{2n}\):
\[{\rm{i}}^{2n}(1 + {\rm{i}} + {\rm{i}}^2 + {\rm{i}}^3) = {\rm{i}}^{2n}(0)\]
Any quantity multiplied by zero is zero.
\[{\rm{i}}^{2n}(0) = 0\]
Therefore, the value of the expression \({\rm{i}}^{2n} + {\rm{i}}^{2n+1} + {\rm{i}}^{2n+2} + {\rm{i}}^{2n+3}\) is \(0\).
General Property: Sum of Four Consecutive Powers of i
This problem demonstrates a general property of the imaginary unit \({\rm{i}}\): the sum of any four consecutive integer powers of \({\rm{i}}\) is always equal to zero. This is because the sum \(1 + {\rm{i}} + {\rm{i}}^2 + {\rm{i}}^3 = 1 + {\rm{i}} - 1 - {\rm{i}} = 0\) always appears as a factor when you group four consecutive powers.
Revision Table: Complex Numbers and Powers of i
| Concept | Description | Property |
|---|---|---|
| Imaginary Unit \({\rm{i}}\) | Defined as the square root of -1. | \({\rm{i}}^2 = -1\) |
| Powers of \({\rm{i}}\) | Repeat in a cycle of 4. | \({\rm{i}}^k = {\rm{i}}^{k \pmod 4}\) (with \({\rm{i}}^0=1\)) |
| Sum of 4 consecutive powers of \({\rm{i}}\) | Adding any four powers with exponents differing by 1, starting from any integer exponent. | \({\rm{i}}^k + {\rm{i}}^{k+1} + {\rm{i}}^{k+2} + {\rm{i}}^{k+3} = 0\) for any integer \(k\). |
Additional Information: Understanding Complex Numbers
Complex numbers extend the real number system by including the imaginary unit \({\rm{i}}\). A complex number is written in the form \(a + b{\rm{i}}\), where \(a\) and \(b\) are real numbers. \(a\) is the real part, and \(b\) is the imaginary part.
Complex numbers are used in many areas of mathematics, physics, and engineering, such as electrical engineering (for analyzing AC circuits) and quantum mechanics.
The property of powers of \({\rm{i}}\) repeating every four terms is crucial for simplifying expressions involving high powers of \({\rm{i}}\) and is often tested in exams focusing on complex numbers.
Important Questions from Complex Numbers
Which one of the following is a square root of \(-\sqrt{-1} \)?
What are the roots of equation-I ?
What is the number of common roots of equation-I and equation-II?
If \(z=\frac{1+i √{3}}{1-i √{3}}\) where i = √-1 then what is the argument of z ?
If z 1and z 2are complex numbers with |z 1| = |z 2|, then which of the following is/are correct?
1. z 1= z 2
2. Real part of z 1= Real part of z 2
3. Imaginary part of z 1= Imaginary part of z 2
Select the correct answer using the code given below:
