Problem 81
Question
In Exercises 79 - 82, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion \( \left(1.02\right)^8 = \left(1 + 0.02\right)^8 = 1 + 8\left(0.02\right) + 28\left(0.02\right)^2 + \cdots \). \( \left(2.99\right)^{12} \)
Step-by-Step Solution
Verified Answer
The approximation of \( (2.99)^{12} \) using the Binomial Theorem is the value we get after calculating the sum: \( (3)^{12} - 12*(3)^{11}*(0.01) + \frac{12*11}{2} *(3)^{10}*(0.01)^2 \)
1Step 1: Rewrite the term for Binomial Expansion
Rewrite \( (2.99)^{12} \) as \( (3 - 0.01)^{12} \) because the binomial expansion is easier to implement when the second term after the plus or minus sign is small.
2Step 2: Apply the Binomial Expansion
The Binomial Theorem states that \( (a+b)^n = a^n + n*a^{n-1}*b + \frac{n*(n-1)}{2!} *a^{n-2}*b^2 + ... \). Use this to expand \( (3 - 0.01)^{12} \): \( (3)^{12} - 12*(3)^{11}*(0.01) + \frac{12*11}{2} *(3)^{10}*(0.01)^2 \). We are only considering the first three terms, as the subsequent terms become very small and will not affect the value up to three decimal places.
3Step 3: Calculate the approximation
Calculate the value from the expanded form: \( (3)^{12} - 12*(3)^{11}*(0.01) + \frac{12*11}{2} *(3)^{10}*(0.01)^2 \) to approximate the value of \( (2.99)^{12} \) up to three decimal places.
Key Concepts
Binomial ExpansionPolynomial ApproximationPower of a Binom
Binomial Expansion
Binomial expansion is a way of expressing a power of a binomial—that is, an expression of the form \( (a+b)^n \)—as a sum of terms involving coefficients and the powers of \( a \) and \( b \) separately. The coefficients of these terms are determined by a sequence known as the binomial coefficients, which can be found in Pascal's Triangle or calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
These coefficients correspond to how many ways you can choose \( k \) elements from a set of \( n \) without regard to order. For instance, the binomial expansion of \( (a+b)^n \) includes terms where \( a \) and \( b \) are raised to different powers, and each term represents a specific combination of \( a \) and \( b \) multiplied together.
To visualize, the expanded form of \( (a+b)^n \) looks like this:
These coefficients correspond to how many ways you can choose \( k \) elements from a set of \( n \) without regard to order. For instance, the binomial expansion of \( (a+b)^n \) includes terms where \( a \) and \( b \) are raised to different powers, and each term represents a specific combination of \( a \) and \( b \) multiplied together.
To visualize, the expanded form of \( (a+b)^n \) looks like this:
- When \( n=2 \) (a square), the expansion is \( a^2 + 2ab + b^2 \).
- When \( n=3 \) (a cube), the expansion is \( a^3 + 3a^2b + 3ab^2 + b^3 \).
- And so on, with the general form being \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \).
Polynomial Approximation
Polynomial approximation is a mathematical method used to estimate values of complex functions by simpler ones, typically polynomials. The idea behind this is similar to using a part of the binomial expansion to approximate a value like \( (2.99)^{12} \). This is particularly helpful when dealing with irrational or transcendental numbers, or for calculating values with a certain precision.
In our example from the textbook, only the first few terms of the binomial expansion are necessary to reach the desired level of accuracy. This is because the later terms in the expansion have decreasing significance due to the progressively smaller power of \( 0.01 \), which is the \( b \) term in our binomial, and the factorial in the binomial coefficients. This concept aligns well with polynomial approximation as it relies on the assumption that a function can be represented as a sum of polynomials, and each added term increases the accuracy of the approximation.
With each additional term, the approximation becomes more precise, but for practical purposes, like when we need an estimate to three decimal places, we can ignore the smaller terms. This approach is a cornerstone of numerical methods and is widely applied across sciences and engineering to simplify complex calculations.
In our example from the textbook, only the first few terms of the binomial expansion are necessary to reach the desired level of accuracy. This is because the later terms in the expansion have decreasing significance due to the progressively smaller power of \( 0.01 \), which is the \( b \) term in our binomial, and the factorial in the binomial coefficients. This concept aligns well with polynomial approximation as it relies on the assumption that a function can be represented as a sum of polynomials, and each added term increases the accuracy of the approximation.
With each additional term, the approximation becomes more precise, but for practical purposes, like when we need an estimate to three decimal places, we can ignore the smaller terms. This approach is a cornerstone of numerical methods and is widely applied across sciences and engineering to simplify complex calculations.
Power of a Binom
The power of a binom refers to raising a binomial expression—comprising two terms, often denoted as \( (a+b) \)—to a certain power, \( n \) in this case. The binomial theorem provides a systematic way to expand such an expression into a sum of terms involving different powers of \( a \) and \( b \) as outlined in the binomial expansion explanation.
When dealing with expressions like \( (2.99)^{12} \), which can be rewritten as \( (3-0.01)^{12} \) to create a binomial where \( a = 3 \) and \( b = -0.01 \), we can see the usefulness of understanding the power of a binom. By applying the binomial theorem, students can find an approximate value of a binom raised to a power without having to perform the entire laborious multiplication process.
By recognizing a pattern and knowing which terms can be omitted for an approximation (typically those of lesser significance as the series progresses), one can quickly estimate the power of a binom to a reasonable degree of accuracy, greatly simplifying problems in algebra, physics, finance, and statistics, among others. This is particularly useful in real-world applications where exact answers may not be necessary, and manageable approximations are sufficient.
When dealing with expressions like \( (2.99)^{12} \), which can be rewritten as \( (3-0.01)^{12} \) to create a binomial where \( a = 3 \) and \( b = -0.01 \), we can see the usefulness of understanding the power of a binom. By applying the binomial theorem, students can find an approximate value of a binom raised to a power without having to perform the entire laborious multiplication process.
By recognizing a pattern and knowing which terms can be omitted for an approximation (typically those of lesser significance as the series progresses), one can quickly estimate the power of a binom to a reasonable degree of accuracy, greatly simplifying problems in algebra, physics, finance, and statistics, among others. This is particularly useful in real-world applications where exact answers may not be necessary, and manageable approximations are sufficient.
Other exercises in this chapter
Problem 80
In Exercises 77-84, simplify the factorial expression. \( \dfrac{10! \cdot 3!}{4! \cdot 6!} \)
View solution Problem 81
In Exercises 79 - 86, solve for \( n \). \( _nP_4 = 10 \cdot _{n - 1} P_3 \)
View solution Problem 81
In Exercises 81 - 85, determine whether the statement is true or false. Justify your answer. If the statement \( p_1 \) is true but the true statement \( P_6 \)
View solution Problem 81
In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{n=0}^{40}2\left(-\dfrac{1}{4}\right)^n \)
View solution