Introduction to Exponential Comparison

When dealing with large numbers and exponential expressions, it can often be difficult to determine which expression is larger. In this article, we will explore how to compare 199^{100}, 200^{100}, and 201^{100}. We will use the binomial theorem and other mathematical techniques to provide a clear and step-by-step solution.

Using the Binomial Theorem

To compare these expressions, we can start by rewriting 201100 using the binomial theorem. The binomial theorem states that for any real number x and positive integer n,

(x y)n Σk0..n (n choose k) * (xn-k) * (yk).

Applying this theorem to our problem, we get:

201100 (200 1)100 Σk0..100 (100 choose k) * (200100-k) * (1k)

This expands to:

200100 100 * 20099 (100 * 99 / 2) * 20098 ... 1

Notice that the first term is exactly 200100, and the remaining terms are all positive additional terms, which means:

201100 200100 additional positive terms

Comparison of 199100 with 200100 and 201100

Now, let's compare 199^{100} with 200^{100} and 201^{100}.

We know that 199

199100 100

However, we also need to consider the additional terms in the expansion of 201100. The additional terms in the binomial expansion are:

100 * 20099 (100 * 99 / 2) * 20098 ... 1

Since these terms are all positive, we can conclude:

201100 > 200100

Combining the inequalities, we get:

199100 100 100

General Approach and Intuition

Considering the general case where n^{100} and (n 1)^{100} , the difference becomes more significant as n increases. At n 0 and n 1, the differences are:

0100 100 100

2100 - 1100 ≈ 1.3 * 1030

3100 - 2100 ≈ 5.2 * 1047

As n increases, the difference only grows larger, indicating that:

199100 100 100

Conclusion

Based on the analysis and the use of the binomial theorem, we can confidently conclude that:

199100 100 100

This result demonstrates the power of exponential growth, showing that small increases in the base can significantly impact the size of the expression when raised to a large power.