Problem 59 - A Complete Resource Book in Mathematics for JEE Main

Question

If \([x]\) denotes the greatest integer less than or equal to \(x\), then \(\left[(6 \sqrt{6}+14)^{2 n+1}\right]\) (A) is an even integer (B) is an odd integer (C) depends on \(n\) (D) none of these

Step-by-Step Solution

Answer

(B) is an odd integer.

Step 1: Understanding the Greatest Integer Function

The function \([x]\) represents the greatest integer less than or equal to \(x\). This means if \(x\) is a non-integer, \([x]\) is the integer part of \(x\). If \(x\) is already an integer, then \([x] = x\).

Step 2: Simplifying the Expression

Consider the expression \((6\sqrt{6}+14)^{2n+1}\). The greatest integer function applies to this expression. Notice that \(2n+1\) indicates the power is always an odd integer regardless of the value of \(n\).

Step 3: Analyzing \\(6\sqrt{6} + 14\\)

Calculate an approximation of \(6\sqrt{6} + 14\). Since \(\sqrt{6} \approx 2.45\), then \(6\sqrt{6} \approx 14.7\). Hence, \(6\sqrt{6} + 14 \approx 28.7\). This shows that \(6\sqrt{6} + 14\) is close to 29 but less than it.

Step 4: Exploring Power Properties

For odd powers \((2n+1)\), \((b)^{2n+1}\) retains the parity (even or odd) of its base when evaluated in the context of greatest integer function. Here, the base is \(6\sqrt{6} + 14\) which approximates to 28.7. Thus, \((28.7)^{2n+1}\) behaves similarly in this context.

Step 5: Determine if Integer is Even or Odd

Since \((6\sqrt{6}+14)\) approximates to 28.7, it is close to 29. Being slightly less than an integer, for large powers, it still resembles 29 when using \([\ ]\), which is odd. Therefore, the greatest integer function applied to this expression yields an odd integer.

Key Concepts

Parity of Integers

Parity is a term used to describe whether a number is even or odd. It is a fundamental concept that helps identify the nature of a number based on its divisibility by 2. If a number is divisible by 2, it is even and if it is not, it is odd. Understanding parity is essential when dealing with integers because:

• Even numbers include 0, 2, 4, 6, etc.
• Odd numbers include 1, 3, 5, 7, etc.

In the context of this exercise, we need to determine whether an integer produced by the greatest integer function is even or odd. As outlined, the expression \( (6 \sqrt{6} + 14)^{2n+1} \) involves odd powers resulting from \(2n+1\).
This power retains the odd nature, reflecting the parity of its approximate base 28.7, which when approximated to the nearest integer, is 29. Hence, the final integer, when rounded down by the greatest integer function, results in an odd parity.

Properties of Functions

Functions have various properties that allow us to predict their behavior. Understanding these properties is crucial to interpreting and solving complex mathematical problems involving functions. The greatest integer function \( [x] \) is particularly notable for its unique attribute where it takes a real number and "steps down" to the nearest whole integer. Here are some key properties related to this exercise:

• The Greatest Integer Function, also called the floor function, rounds down a real number to the largest integer less than or equal to it.
• In this problem, \( [ (6 \sqrt{6} + 14)^{2n+1} ] \) indicates that no matter the fraction component after taking the power, the function disregards it.

Due to the function's properties, it aids in determining whether a resulting integer is even or odd by focusing solely on the largest completed integer, ensuring fractional values don't affect the result.

Approximation Techniques

Approximation techniques play a vital role when precise calculations are difficult or unnecessary. When dealing with expressions that involve irrational numbers like \(6 \sqrt{6}\ \), it often becomes essential to approximate them to simplify calculations. Here's how it's done in this exercise:

• Calculate \( \sqrt{6} \) which approximately equals 2.45.
• Then, compute \( 6 \times 2.45 \approx 14.7 \).
• Add this to 14 to get \( 28.7 \).

This approximation shows that the value is just shy of 29, but remains less. Applying the greatest integer function drops this down to 28, but when raised to an odd power, retains fractional properties that push its integer floor approximation behavior subtly closer to 29, staying influenced by an odd characteristic.
Such approximation is handy for estimations, especially in exams or scenarios requiring quick verification.