Problem 38 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(S(n)=i^{n}+i^{-n}\), where \(i=\sqrt{-1}\) and \(n\) is a positive integer, then the total number of distinct values of \(S(n)\) is (A) 1 (B) 2 (C) 3 (D) 4

Step-by-Step Solution

Answer

The answer is (C) 3.

Step 1: Understanding Complex Powers

The exercise asks for the distinct values of the expression \( S(n) = i^n + i^{-n} \), where \( i = \sqrt{-1}\). The powers of \( i \) cycle every four steps: \( i = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \), then it repeats. Knowing this allows us to determine \( i^{-n} \) given \( i^n \).

Step 2: Evaluate Expression for n=1

For \( n=1 \), we have \( S(1) = i + i^{-1} \). To find \( i^{-1} \), consider \( i \cdot i^{-1} = 1 \), leading to \( i^{-1} = -i \). Thus, \( S(1) = i + (-i) = 0 \).

Step 3: Evaluate Expression for n=2

For \( n=2 \), we have \( S(2) = i^2 + i^{-2} = -1 + (-1) = -2 \).

Step 4: Evaluate Expression for n=3

For \( n=3 \), we have \( S(3) = i^3 + i^{-3} = -i + i = 0 \).

Step 5: Evaluate Expression for n=4

For \( n=4 \), we have \( S(4) = i^4 + i^{-4} = 1 + 1 = 2 \).

Step 6: Identify Distinct Values

The distinct values from the computations are \( 0, -2, \) and \( 2 \). Thus, there are 3 distinct values of \( S(n) \).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the field of complex numbers. It is defined as the square root of negative one: \(i = \sqrt{-1}\). This unique property allows us to extend the real number system to include solutions to equations involving square roots of negative numbers.

By itself, \(i\) isn't a real number, but when combined with real numbers, it forms what we call complex numbers. A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(b\) is the imaginary part.

Understanding \(i\) is crucial because it forms the basis for explaining more advanced mathematical concepts like complex powers and through cycles of powers, which we'll discuss next.

Complex Powers

Complex powers of \(i\) exhibit a fascinating cyclical nature due to its properties. When \(i\) is raised to successive powers, it cycles through four distinct outcomes:

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

This cycle repeats every four terms. The cycle restarts with \(i^5 = i\), \(i^6 = -1\), and so on. This pattern simplifies computations involving powers of \(i\), since you can determine the power \(i^n\) using modular arithmetic (i.e., find the remainder of the division of \(n\) by 4).

This cyclical behavior is especially useful in exercises like the one given, where for any integer \(n\), we can easily determine both \(i^n\) and its reciprocal \(i^{-n}\). We leverage these cyclical properties to compute values efficiently, as shown in the original solution steps.

Distinct Values

The expression \(S(n) = i^n + i^{-n}\) gives us an opportunity to explore how the cyclical nature of complex powers leads to distinct values. We have found that each value of \(n\) results in specific combinations of \(i^n\) and its reciprocal, \(i^{-n}\).

Let's break down how these combinations lead to distinct results:

• When \(n = 1\) or \(n = 3\), the expression results in \(S(n) = 0\).
• For \(n = 2\), \(S(n)\) equals \(-2\).
• At \(n = 4\), \(S(n)\) becomes \(2\).

These outcomes show that the distinct values of \(S(n)\) are \(0, -2,\) and \(2\). Thus, the total number of distinct values is 3. Recognizing these patterns is beneficial for quickly solving similar problems without exhaustive calculations.