Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

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 total number of distinct values of \(S(n)\) is 3.

Step 1: Simplify the expression for S(n)

Recall that the imaginary unit is defined by \(i^2 = -1\). We need to evaluate \(S(n) = i^n + i^{-n}\). Let's compute the first few values of \(S(n)\) to find a pattern.

Step 2: Evaluate powers of i

For \(n = 1\), we have \(i^1 = i\) and \(i^{-1} = -i\), so \(S(1) = i - i = 0\). For \(n = 2\), \(i^2 = -1\) and \(i^{-2} = -1\), therefore \(S(2) = -1 - 1 = -2\). For \(n = 3\), \(i^3 = -i\) and \(i^{-3} = i\), so \(S(3) = -i + i = 0\). For \(n = 4\), \(i^4 = 1\) and \(i^{-4} = 1\), thus \(S(4) = 1 + 1 = 2\).

Step 3: Observe the pattern

The values we obtained from \(S(n)\) are: \(0, -2, 0, 2\). Notice that this pattern repeats as \((0, -2, 0, 2)\) periodically because the powers of \(i\) repeat every four terms: \(i, -1, -i, 1\).

Step 4: Identify distinct values

The distinct values of \(S(n)\) are \(0, -2, 2\). Therefore, there are three distinct values that \(S(n)\) can take.

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept when it comes to complex numbers. It is defined as the square root of \(-1\), which means that \(i^2 = -1\). This definition allows for the extension of real numbers to complex numbers where the real part and the imaginary part together form a complex plane.
Complex numbers are typically written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary unit \(i\) enables calculations that would not be possible with just real numbers, and it introduces a new dimension of numbers that can be manipulated just like real numbers.

• It helps solve equations that do not have real solutions, like \(x^2 = -1\).
• Plays a critical role in fields such as engineering, physics, and applied mathematics.

With the concept of \(i\), we can explore various mathematical phenomena, including rotations in the complex plane, which represent the powers of \(i\).

Powers of Complex Numbers

The powers of the imaginary unit \(i\) exhibit a repeating cycle, which simplifies calculations involving \(i\). Here’s how they progress:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• Then the cycle repeats: \(i^5 = i\), \(i^6 = -1\), and so forth.

This cycle repeats every four powers, making it convenient to predict the value of any power of \(i\) by evaluating the exponent modulo 4. For instance, \(i^{11} = i^{11 \mod 4} = i^3 = -i\).
Knowing this pattern is crucial for problems that involve higher powers of \(i\), as it allows you to quickly determine the behavior of expressions like \(i^n\) without calculating each power individually. It also aids in understanding sequences and transformations in the context of complex arithmetic.

Patterns in Sequences

When dealing with sequences involving complex numbers, recognizing repeating patterns is essential. The problem we examine here involves the sequence \(S(n) = i^n + i^{-n}\). By assessing each power of \(i\), we observe that the sequence of powers \(i, -1, -i, 1\) and their reciprocals repeat every four terms. Thus, it affects the sequence \(S(n)\) as follows:

• \(S(1) = i - i = 0\)
• \(S(2) = -1 - 1 = -2\)
• \(S(3) = -i + i = 0\)
• \(S(4) = 1 + 1 = 2\)

Analyzing these terms shows a pattern: \(0, -2, 0, 2\), repeating every four terms. The distinct values in this sequence are 0, -2, and 2.
By recognizing these repeating elements, students can easily predict the value of the sequence \(S(n)\) for any positive integer \(n\), and understand the behavior of complex number sequences in broader mathematical contexts.