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

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

The answer is 2.

Step 1: Understanding the Expression

We need to find an integer greater than \( (1 + 0.0001)^{1000} \). The expression involves a small amount raised to a large power, which suggests using an approximation method.

Step 2: Using Binomial Approximation

To approximate \( (1 + x)^n \) when \( x \) is small, we can use the binomial theorem. For small \( x \), \( (1 + x)^n \approx 1 + nx \). Here, \( x = 0.0001 \) and \( n = 1000 \), so the approximation is \( 1 + 1000 \times 0.0001 = 1 + 0.1 = 1.1 \).

Step 3: Finding the Integer Just Greater

From the approximation, we know that \( (1 + 0.0001)^{1000} \approx 1.1 \). The positive integer greater than 1.1 is 2.

Key Concepts

Binomial Approximation

To handle the expression \( (1 + 0.0001)^{1000}\), we use the binomial approximation as it simplifies complex expressions involving powers of small numbers. The binomial theorem provides a convenient way to approximate \( (1 + x)^n\) when the absolute value of \( x \) is small and \( n\) is large. In essence, the approximation states that:

• \( (1 + x)^n \approx 1 + nx \)

This formula holds true because the higher-order terms of the binomial expansion become insignificant as \( x \) diminishes in size. Applying this to our problem with \( x = 0.0001 \) and \( n = 1000 \), we simplify the expression to \( 1 + 1000 imes 0.0001 = 1.1 \).
This approximately gives us 1.1 as a result. The binomial approximation is a quick way to assess values without needing full calculations, and is particularly useful in various fields including statistics, physics, and computational mathematics.

Positive Integer Approximation

In mathematical problems, finding the closest integer can sometimes be more practical than an exact decimal result. For the expression \( (1 + 0.0001)^{1000} \), our goal is to approximate the resulting value to a positive integer. Based on the binomial approximation result of 1.1, the next task is to identify the smallest positive integer greater than this value.
Since 1.1 is slightly more than a whole number, rounding up gives us a positive integer of 2. Here, understanding the proximity of our approximation to integers helps in determining the final value required. In contexts such as elementary number theory and certain computational applications, approximate integers suffice for the decision-making process.

Exponential Growth Analysis

Exponential growth involves rapid increases in quantity over time, governed by constant multiplicative factors. In \( (1 + 0.0001)^{1000} \), we explore how even a minuscule base factor like 0.0001 alters the outcome over large exponents (1000, in this case).
The tiny factor, when multiplied over 1000 growth cycles, results in the overall product becoming significantly larger than the initial quantity. This concept of exponential escalation is fundamental in analyzing phenomena such as population growth, compound interest in finance, and radioactive decay. By looking at \( (1 + 0.0001)^{1000} \), we begin to appreciate how small percentage increases significantly affect exponential outcomes over substantial periods or cycles. This understanding is vital for applications where prediction over time is crucial, as exponential scaling can lead to unexpectedly large results.