Problem 71 - Contemporary Precalculus

Question

Use the Binomial Theorem to show that \(1.001^{1000}>2\) [Hint: Write 1.001 as a sum.]

Step-by-Step Solution

Answer

Question: Show that \(1.001^{1000} > 2\) using the Binomial Theorem. Answer: Using the Binomial Theorem and analyzing the first three terms of the expansion, we found that the sum of all terms in our expansion is greater than 2, demonstrating that \(1.001^{1000} > 2\).

Step 1: Expanding the Expression using Binomial Theorem

Using the binomial theorem, we can expand \((1 + 0.001)^{1000}\) as: $$ (1 + 0.001)^{1000} = \binom{1000}{0}1^{1000}0.001^0 + \binom{1000}{1}1^{999}0.001^1 + \binom{1000}{2}1^{998}0.001^2 + \dots + \binom{1000}{1000}1^00.001^{1000} $$ Since \(1^{n}=1\) for any integer \(n\), we can simplify the expression: $$ (1 + 0.001)^{1000} = \binom{1000}{0} + \binom{1000}{1}0.001 + \binom{1000}{2}0.001^2 + \dots + \binom{1000}{1000}0.001^{1000} $$

Step 2: Lower Bound of the Expression

The first three terms of the expansion are: $$ \binom{1000}{0}=1, \quad \binom{1000}{1}0.001 = 1, \quad \text{and} \quad \binom{1000}{2}0.001^2 = \frac{1000 \times 999}{2} 0.001^2 $$ In this exercise, we only need to show that \(1.001^{1000}>2\). It is clear that the first two terms of the expansion add to 2: $$ \binom{1000}{0} + \binom{1000}{1}0.001 = 1 + 1 = 2 $$ Since the remaining terms are all positive (binomial coefficients are always positive and the powers of 0.001 are also positive), the sum of all terms in our expansion must be greater than 2: $$ 1.001^{1000} = \binom{1000}{0} + \binom{1000}{1}0.001 + \binom{1000}{2}0.001^2 + \dots + \binom{1000}{1000}0.001^{1000} > 2 $$ Therefore, using the Binomial Theorem, we've shown that \(1.001^{1000} > 2\).

Key Concepts

Binomial Expansion

Understanding the concept of binomial expansion is critical for grasping the fundamentals of algebra and calculus. Put simply, binomial expansion enables us to express a binomial expression raised to a power as a sum of terms involving coefficients and the individual terms of the binomial. When we refer to a binomial, we're talking about an algebraic expression that contains two terms, such as \(a+b\).

The Binomial Theorem provides a formula for expanding expressions of the form \( (a+b)^n\), where \(n\) is a non-negative integer. This theorem states that:
\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Here, \(\binom{n}{k}\) are the binomial coefficients, and the summation symbol \(\sum\) denotes the sum of all terms from \(k=0\) to \(k=n\). Each term in the series is a product of a binomial coefficient, a power of \(a\), and a power of \(b\).

In the provided exercise, we apply the binomial expansion to the expression \(1.001^{1000}\), which is the same as \( (1+0.001)^{1000}\). This helps in accurately calculating the result, and as shown, even the first few terms of the expansion can provide insight into the minimum value of the expression. The Binomial Theorem is therefore not just a theoretical construct but a powerful tool for evaluating large exponents with precision.

Binomial Coefficients

The binomial coefficients are arguably the stars of the binomial expansion. They are the numbers that appear as multipliers in the expansion of a binomial expression, and they hold significant combinatorial meaning.

Binomial coefficients are denoted by \( \binom{n}{k} \), which is read as \