Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The positive integer which is just greater than \((1+0.0001)^{1000}\) is (A) 3 (B) 4 (C) 5 (D) 2

Step-by-Step Solution

Answer

2

Step 1: Understand the Expression

The expression we need to evaluate is \((1 + 0.0001)^{1000}\). This involves calculating a power of a number slightly greater than 1.

Step 2: Approximate Using the Binomial Theorem

For small \( x \), \( (1+x)^n \approx 1+nx \) when \( x \) is very small. Here, \( x = 0.0001 \) and \( n = 1000 \). Thus, we approximate: \((1 + 0.0001)^{1000} \approx 1 + 1000 \cdot 0.0001 = 1 + 0.1 = 1.1.\)

Step 3: Determine the Nearest Integer Greater Than the Approximation

The calculation yields \( 1 + 0.1 = 1.1 \). The smallest integer greater than 1.1 is 2.

Step 4: Verify Feasibility of Other Options

Check that the approximation is reasonable: for \( (1 + 0.0001)^{1000} \) to be closer to 3, 4, or 5, the approximation would have needed to be significantly higher than 1.1, which is not the case here.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation where a number, called the base, is multiplied by itself a certain number of times. This number of times is given by the power, or the exponent. For instance, in the expression \( (1 + 0.0001)^{1000} \), 1.0001 is the base, and 1000 is the exponent.

• The process of exponentiation can lead to rapid growth of the result, depending on how large the exponent is.
• When the base is a number slightly greater than 1, as in the exercise, the growth can be modest, especially for large exponents.

In this exercise, the exponentiation of 1.0001 to the power of 1000 yields a value that's a bit larger than 1. However, the approximate growth is what we estimate using approximation techniques like the Binomial Theorem.

Approximation

Approximation is a technique used to find a value that is close enough to the right answer, often within a tolerance that is acceptable for practical purposes. In mathematics, when dealing with complex calculations, approximations can simplify the process significantly.
For example, using the Binomial Theorem, we approximate the value of expressions like \((1+x)^n\) when \(x\) is small. The theorem states that:

• \((1+x)^n \approx 1+nx\) when \(x\) is very close to zero.
• This simplification works well when \(n\) is large and \(x\) is very small, as it is in our exercise where \(x = 0.0001\) and \(n = 1000\).

Thus, the expression \( (1 + 0.0001)^{1000} \) was approximated to \(1 + 0.1 = 1.1\). This approximation provides a quick way to foresee how close or how much larger the actual value is compared to a simple number like 1.1.

Positive Integer

A positive integer is any whole number greater than zero. This set of numbers includes 1, 2, 3, and so on. In mathematical contexts, especially when working with approximations and series, identifying a positive integer that's nearest or convenient to an estimated value is a common task.

• In our problem, after approximation, the value was approximately 1.1.
• The task was to determine which positive integer is just greater than 1.1.

Among positive integers like 1, 2, 3, and 4, the smallest number greater than 1.1 is 2. Therefore, understanding and working with positive integers is essential in determining which of these values best fits in the context of this calculation.