If i stands for √(-1), then, for positive integers n1, n2, the value of the expression (1+i)^n1+

step 1 In this question we have an expression 1 plus iota to the power n1 plus 1 plus iota cube to the

If i stands for √(-1), then, for positive integers n1, n2,...

If i stands for $\sqrt{-1}$, then, for positive integers $\mathrm{n}_{1}, \mathrm{n}_{2}$, the value of the expression $(1+\mathrm{i})^{\mathrm{n}_{1}}+\left(1+\mathrm{i}^{3}\right)^{\mathrm{n}_{1}}+\left(1+\mathrm{i}^{5}\right)^{\mathrm{n}_{2}}+\left(1+\mathrm{i}^{7}\right)^{\mathrm{n}_{2}}$ is a real number if and only if (a) $\mathrm{n}_{1}=\mathrm{n}_{2}+1$ (b) $n_{1}=n_{2}-1$ (c) $\mathrm{n}_{1}=\mathrm{n}$, (d) $\mathrm{n}$, and $\mathrm{n}_{2}$ can be any positive integers

If i stands for $\sqrt{-1}$, then, for positive integers $\mathrm{n}_{1}, \mathrm{n}_{2}$, the value of the expression $(1+\mathrm{i})^{\mathrm{n}_{1}}+\left(1+\mathrm{i}^{3}\right)^{\mathrm{n}_{1}}+\left(1+\mathrm{i}^{5}\right)^{\mathrm{n}_{2}}+\left(1+\mathrm{i}^{7}\right)^{\mathrm{n}_{2}}$
is a real number if and only if
(a) $\mathrm{n}_{1}=\mathrm{n}_{2}+1$
(b) $n_{1}=n_{2}-1$
(c) $\mathrm{n}_{1}=\mathrm{n}$,
(d) $\mathrm{n}$, and $\mathrm{n}_{2}$ can be any positive integers

Key Concepts

De Moivre's Theorem

De Moivre's theorem states that raising a complex number (expressed in polar form) to an integer power n involves raising its modulus to the n-th power and multiplying its angle by n. This theorem is instrumental in simplifying expressions involving powers of complex numbers and understanding their behavior.

Polar Form and Euler's Formula

Representing complex numbers in polar form, where a number is expressed as r·exp(i?) (with r as the magnitude and ? as the argument), allows for easier manipulation when raising numbers to a power. Euler's formula connects this representation with trigonometric functions, which is crucial for applying trigonometric identities in computations.

Conditions for a Real Sum of Complex Terms

For a sum of complex numbers to be real, the imaginary parts must cancel out. Recognizing that pairing conjugate expressions or ensuring their arguments lead to cancellation of imaginary components is central to determining the conditions under which a complex sum yields a real number.

Imaginary Unit and Its Cyclic Powers

The imaginary unit, typically represented by i (with i² = -1), exhibits a cyclic pattern in its powers due to the periodic nature of complex multiplication. This cyclicity (with period 4) allows for the simplification of expressions involving higher powers of i by reducing the exponent modulo 4.

Complex Conjugates

A complex conjugate reverses the sign of the imaginary part while keeping the real part unchanged. When a complex number and its conjugate are added together, the imaginary parts cancel, resulting in a real number. This property is key to determining under what conditions sums of complex expressions can be real.

Transcript

Please give Ace some feedback