Math Problem Statement

(1 + i)^n + (1 - i)^n = 2 * (n+1)/2 * cos(nπ/4)

Solution

The expression in the image is as follows:

(1+i)n+(1i)n=2n+12cos(nπ4)(1 + i)^n + (1 - i)^n = 2 \cdot \frac{n+1}{2} \cdot \cos\left(\frac{n\pi}{4}\right)

This is a combination of complex numbers and trigonometric functions. To analyze it:

Key Components:

  1. (1+i)n(1 + i)^n and (1i)n(1 - i)^n: These represent powers of complex numbers. We can simplify them using De Moivre's theorem, which expresses powers of complex numbers in terms of trigonometric functions.

  2. cos(nπ4)\cos\left(\frac{n\pi}{4}\right): This is a cosine function evaluated at angles that are multiples of π4\frac{\pi}{4}, which corresponds to common angles on the unit circle like 4545^\circ, 9090^\circ, etc.

Simplification Strategy:

  • Using De Moivre's theorem: (1+i)=2(cosπ4+isinπ4)(1+i) = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) So: (1+i)n=(2)n(cos(nπ4)+isin(nπ4))(1 + i)^n = \left(\sqrt{2}\right)^n \cdot \left(\cos\left(\frac{n\pi}{4}\right) + i\sin\left(\frac{n\pi}{4}\right)\right) Similarly, for (1i)n(1 - i)^n: (1i)n=(2)n(cos(nπ4)isin(nπ4))(1 - i)^n = \left(\sqrt{2}\right)^n \cdot \left(\cos\left(\frac{n\pi}{4}\right) - i\sin\left(\frac{n\pi}{4}\right)\right)

  • Adding (1+i)n(1+i)^n and (1i)n(1-i)^n, the imaginary parts cancel, and we are left with: (1+i)n+(1i)n=2(2)ncos(nπ4)(1 + i)^n + (1 - i)^n = 2 \cdot \left(\sqrt{2}\right)^n \cdot \cos\left(\frac{n\pi}{4}\right)

From the equation in the image: 2(2)ncos(nπ4)=2n+12cos(nπ4)2 \cdot \left(\sqrt{2}\right)^n \cdot \cos\left(\frac{n\pi}{4}\right) = 2 \cdot \frac{n+1}{2} \cdot \cos\left(\frac{n\pi}{4}\right)

Canceling out the cosine terms and simplifying: (2)n=n+12\left(\sqrt{2}\right)^n = \frac{n+1}{2}

Thus, this is the resulting equation that relates the power of 2\sqrt{2} with nn.

Would you like further details on solving this equation or have any other questions?

5 Related Questions:

  1. How does De Moivre's theorem work for complex numbers?
  2. What are the geometric interpretations of (1+i)n(1+i)^n on the complex plane?
  3. Can this equation help find specific values of nn?
  4. How can we graph the cosine function for multiples of π4\frac{\pi}{4}?
  5. How is the equation (2)n=n+12\left(\sqrt{2}\right)^n = \frac{n+1}{2} solved for nn?

Tip:

When dealing with powers of complex numbers, using polar forms and De Moivre's theorem often simplifies calculations dramatically.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Trigonometry
De Moivre's Theorem

Formulas

(1 + i)^n + (1 - i)^n = 2 * (n+1)/2 * cos(nπ/4)
(1 + i)^n = (√2)^n * (cos(nπ/4) + i sin(nπ/4))
(1 - i)^n = (√2)^n * (cos(nπ/4) - i sin(nπ/4))

Theorems

De Moivre's Theorem

Suitable Grade Level

Undergraduate Mathematics