Problem 45 - Algebra and Trigonometry

Question

Show that \((1.01)^{100}>2\) [Hint: Note that \((1.01)^{100}=(1+0.01)^{100}\) and use the Binomial Theorem to show that the sum of the first two terms of the expansion is greater than 2.1

Step-by-Step Solution

Answer

The sum of additional terms from the expansion of \((1+0.01)^{100}\) shows \((1.01)^{100} > 2\).

Step 1: Understand the Goal

We need to show that \((1.01)^{100} > 2\). This involves understanding the expression in terms of known expansions or approximations.

Step 2: Recognize the Binomial Expansion

Use the Binomial Theorem to expand \((1.01)^{100} = (1 + 0.01)^{100}\). The theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = 1\), \(b = 0.01\), and \(n = 100\).

Step 3: Expand Using First Few Terms

According to the Binomial Theorem, the first few terms of the expansion are:\[1^n + \binom{100}{1}(1^{99})(0.01)^1 + \binom{100}{2}(1^{98})(0.01)^2 + \ldots\]Let's calculate these terms:\[1 + 100\times 0.01 + \frac{100 \times 99}{2} \times (0.01)^2 + \ldots\]

Step 4: Calculate the First Two Terms

Calculate the value of the first two terms:- The first term is 1.- The second term is \(100 \times 0.01 = 1\).- Sum: \(1 + 1 = 2\). This result is not greater than 2.1, so we need to check more terms.

Step 5: Include the Third Term

Now calculate the third term:- The third term is \(\frac{100 \times 99}{2} \times (0.01)^2 = 49.5 \times 0.0001 = 0.00495\).- Sum of the first three terms: \(1 + 1 + 0.00495 \approx 2.00495\).- This sum still does not show \((1.01)^{100} > 2.1\) directly, so consider even more terms if necessary.

Step 6: Compute Additional Terms (Fourth Term)

Ironically, it was incorrect to directly aim for 2.1, instead use the correct computation:- Fourth term: \(\frac{100 \times 99 \times 98}{6} \times (0.01)^3 = 1617 \times 0.000001 = 0.001617\).- Sum: \(2.00495 + 0.001617 \approx 2.006567\), not decisive alone.

Step 7: Consecutive Correction

We mistake aiming directly for a total which should immediately verify the sum rather leveraging known factored bound for choosing easier calculus direct result outcome - seeing the sum worked converge exponential performance quickly confirmed exceeds near immediate convergence additional functioning.

Key Concepts

Exponential Growth

Exponential growth is a powerful mathematical concept that describes how quantities increase rapidly over time. When a quantity grows exponentially, it means that the rate of growth is proportional to the current size. This leads to a very rapid increase.
Exponential growth is expressed mathematically as:

• \(y = a(1 + r)^t\)
• where \(y\) is the final amount, \(a\) is the initial amount, \(r\) is the rate of growth, and \(t\) is the time period.

In the context of the exercise, we look at the exponential expression \((1.01)^{100}\). This expression signifies repeated multiplication over time, which is a hallmark of exponential growth. It helps us understand not only rapid increase but also how small incremental rates (like 1% here) compounded over many iterations can lead to significant outcomes.
Understanding exponential growth is crucial as it applies to myriad real-life scenarios like finance, population dynamics, and technology proliferation.

Algebraic Proofs

Algebraic proofs employ a series of logical deductions to establish the truth of a given statement using algebraic principles. In the exercise, we set out to prove that \((1.01)^{100} > 2\) using the Binomial Theorem.
The Binomial Theorem enables us to expand algebraic expressions like \((a + b)^n\) into a sum of terms involving coefficients (binomial coefficients), powers of \(a\), and powers of \(b\). This makes it feasible to compute powers of sums in a manageable manner.
For the proof, we use the first few terms in the expansion:

• The first term is \(1^n = 1\).
• The second term involves the first-order binomial coefficient: \(100 \times 0.01 = 1\).
• The third term includes the quadratic binomial coefficient: \(\frac{100 \times 99}{2} \times (0.01)^2 = 0.00495\).

Algebraic proofs like these rely on logical steps and calculations to confirm conclusions. The approach typically involves starting from the known, applying theoretical tools (like the binomial theorem), and concluding with a verified statement that addresses the problem at hand.

Mathematical Inequalities

Mathematical inequalities are expressions that describe the relative size or order of two values. Inequalities are a fundamental part of mathematics, opening the door to understanding limits, ranges, and optimum scenarios.
In this exercise, we aim to prove that

• \((1.01)^{100} > 2\).

This is an example of a mathematical inequality where we establish that one value exceeds another. Using the binomial expansion, we calculated successive terms, establishing step by step that the sum grows beyond 2, leveraging the concept of a growing series.
This concept aids in showcasing real-world comparisons where exact equality isn't required, and a relational comparison (greater, lesser, or equal) provides necessary insights. Understanding inequalities is crucial as it frequently appears in optimization problems, risk assessments, and various theoretical and applied sciences where bounds, constraints, or approximations are necessary.