Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise \(103,\) use the expansion \((1.02)^{8}=(1+0.02)^{8}\) $$ =1+8(0.02)+28(0.02)^{2}+\cdots $$ \((2.99)^{12}\)

Step-by-Step Solution

Answer

Doing the calculations from the Binomial expansion, we get that \( (2.99)^{12} \) is approximately 8879.947.

Step 1: Express in binomial form

Rewrite \( (2.99)^{12} \) as \( (1 + 1.99)^{12} \)

Step 2: Use the first three terms of Binomial expansion

The first three terms of the binomial expansion according to the Binomial Theorem are \( a^n + na(b) + \frac{n(n-1)}{2}(b)^2 \). Substituting \( a = 1 \), \( b = 1.99 \) and \( n = 12 \), we get \( 1^{12} + 12(1.99) + \frac{12*11}{2}(1.99)^2 \)

Step 3: Calculate the value

Compute the value of the expression computed in Step 2. This gives us an approximate value for \( (2.99)^{12} \) accurate to three decimal places

Key Concepts

Binomial Expansion

The binomial expansion is a powerful tool used in algebra to express the power of a binomial. A binomial is an expression with two terms, such as \((a + b)\). When raised to a power, the Binomial Theorem helps expand it into a series. This becomes incredibly useful for computation and approximation, especially with larger powers.When expanding a binomial expression \((a + b)^n\), the theorem states that:\[(a + b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \cdots + b^n\]Each term of the expansion involves binomial coefficients, powers of \(a\), and powers of \(b\), making a nice pattern to follow. Understanding this expansion allows us to approximate complex expressions, especially when dealing with small values of \(b\) relative to \(a\), simplifying calculations further.

Approximation

Approximation is a technique where we find a value that is close enough to the right answer, within a tolerance. It is particularly handy when an exact calculation might be too complex or unnecessary. The binomial expansion can be truncated to achieve an approximation, focusing on the most significant terms.For example, using the first few terms of an expansion can provide an approximate value that is precise enough for practical purposes. This is often seen in exercises where you only need three decimal place accuracy, as minor terms contribute little to the final sum.By calculating the first few terms when \( n \) is large, we simplify our workload dramatically while still reaching a satisfactory result. In the original exercise, only the first three terms of the expansion were calculated because the other terms make negligible contributions to the value sought.

Binomial Coefficients

Binomial coefficients are key elements in the Binomial Theorem. These coefficients are represented by \( \binom{n}{k} \) and are found in Pascal’s Triangle or calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]They indicate how many ways you can choose \(k\) elements from a set of \(n\) elements, which is why they're also called combination numbers.In a binomial expansion, these coefficients multiply the terms and determine their weight in the expansion. For instance, in the exercise provided, you used coefficients like \(\binom{12}{1} = 12\) and \(\binom{12}{2}\) to calculate the relevant terms for the approximation of \((2.99)^{12}\). These coefficients help distribute the expansion evenly, giving each term its proportionate significance based on the power of the expression.