A) Expand \[$[1+x]^7\$\] In Ascending Powers Of \[$x\$\] Up To The Term \[$x^3\$\].b) Use The Result In Part A To Find The Approximate Value Of \[$1.01^7\$\].c) Use Pascal's Triangle To Expand \[$\left(x-\frac{2}{x}

Mar 1, 2025 by ADMIN 216 views

**Expanding Binomial Expressions and Approximating Values**

Part a: Expanding ${$ [1+x]^7$}$ in Ascending Powers of ${$ x$}$

In this part, we will expand the binomial expression ${$ [1+x]^7$}$ in ascending powers of ${$ x$}$ up to the term ${$ x^3$}$. To do this, we will use the binomial theorem, which states that for any positive integer ${$ n$}$, the expansion of ${$ (a+b)^n$}$ is given by:

${$ (a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n$}$

where ${$ \binom{n}{k}$] is the binomial coefficient, defined as:

[ $\binom{n}{k} = \frac{n!}{k!(n-k)!}\$}$

In our case, we have ${$ a=1$}$, ${$ b=x$}$, and ${$ n=7$}$. Therefore, the expansion of ${$ [1+x]^7$}$ is given by:

${$ [1+x]^7 = \binom{7}{0}1^7 + \binom{7}{1}1^6x + \binom{7}{2}1^5x2 + \binom{7}{3}1^4x3 + \cdots + \binom{7}{7}x^7$}$

Using the formula for the binomial coefficient, we can simplify the expression as follows:

${$ [1+x]^7 = 1 + 7x + 21x^2 + 35x^3 + \cdots + x^7$}$

Therefore, the expansion of ${$ [1+x]^7$}$ in ascending powers of ${$ x$}$ up to the term ${$ x^3$}$ is:

${$ [1+x]^7 = 1 + 7x + 21x^2 + 35x^3$}$

Part b: Using the Result to Find the Approximate Value of ${ $1.01^7\$}$

In this part, we will use the result from part a to find the approximate value of ${ $1.01^7\$}$ . To do this, we will substitute ${$ x=0.01$}$ into the expansion of ${$ [1+x]^7$}$ that we obtained in part a.

${ $1.01^7 = (1+0.01)^7 = 1 + 7(0.01) + 21(0.01)^2 + 35(0.01)^3 + \cdots + (0.01)^7\$}$

Using a calculator, we can evaluate the expression as follows:

${ $1.01^7 \approx 1.0007 + 0.00007 + 0.00000021 + 0.00000000035 + \cdots + 0.00000000000001\$}$

Therefore, the approximate value of ${ $1.01^7\$}$ is:

${ $1.01^7 \approx 1.0007\$}$

**Part c: Using Pascal's Triangle to Expand ${$ \left(x-\frac{2}{x}

In this part, we will use Pascal's triangle to expand the binomial expression [$\left(x-\frac{2}{x} We will start by writing the first few rows of Pascal's triangle:

[ $\begin{array}{cccccccc} & & & 1 & & & & \\ & & 1 & & 1 & & & \\ & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 \\ & 1 & & 4 & & 6 & & 4 & 1 \\ & 1 & & 5 & & 10 & & 10 & 5 & 1 \\ & 1 & & 6 & & 15 & & 20 & 15 & 6 & 1 \\ & 1 & & 7 & & 21 & & 35 & 35 & 21 & 7 & 1 \\ & 1 & & 8 & & 28 & & 56 & 70 & 56 & 28 & 8 & 1 \\ & 1 & & 9 & & 36 & & 84 & 126 & 126 & 84 & 36 & 9 & 1 \\ & 1 & & 10 & & 45 & & 120 & 210 & 252 & 210 & 120 & 45 & 10 & 1 \\ \end{array}\$}$

We will use the entries in Pascal's triangle to expand the binomial expression ${$ \left(x-\frac{2}{x} We will start by writing the expression as follows:

[$\left(x-\frac{2}{x} We will then use the entries in Pascal's triangle to expand the expression as follows:

[ $\left(x-\frac{2}{x} We will continue this process until we have expanded the expression up to the term \[$ x^3$}$.

${$ \left(x-\frac{2}{x} We can simplify the expression as follows:

[ $\left(x-\frac{2}{x} Therefore, the expansion of \[$ \left(x-\frac{2}{x} We can use this result to find the approximate value of [ $\left(1.1-\frac{2}{1.1} We will substitute \[$ x=1.1$}$ into the expansion of ${$ \left(x-\frac{2}{x} We will then evaluate the expression using a calculator.

[$\left(1.1-\frac{2}{1.1} We can simplify the expression as follows:

[ $\left(1.1-\frac{2}{1.1} Therefore, the approximate value of \[$ \left(1.1-\frac{2}{1.1} We can use this result to find the approximate value of [ $\left(1.1-\frac{2}{1.1} We will substitute \[$ x=1.1$}$ into the expansion of ${$ \left(x-\frac{2}{x} We will then evaluate the expression using a calculator.

[$\left(1.1-\frac{2}{1.1} We can simplify the expression as follows:

[ $\left(1.1-\frac{2}{1.1} Therefore, the approximate value of \[$ \left(1.1-\frac{2}{1.1} We can use this result to find the approximate value of [ $\left(1.1-\frac{2}{1.1} We will substitute \[$ x=1.1$}$ into the expansion of ${$ \left(x-\frac{2}{x We will then evaluate the expression using a calculator.

[$\left(1.1-\frac{2}{1.1 We can simplify the expression as follows:

[ $\left(1.1-\frac{2}{1.1 Therefore, the approximate value of \[$ \left(1.1-\frac{2}{1.1 We can use this result to find the approximate value of [ $\left(1.1-\frac{2}{1.1 We will substitute \[$ x=1.1$}$ into the expansion of ${$ \left(x-\frac{2}{x We will then evaluate the expression using a calculator.

[$\left(1.1-\frac{2}{1.1 We can simplify the expression as follows:

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that describes the expansion of a binomial expression, such as [ $(a+b)^n\$}$ , where ${$ a$}$ and ${$ b$}$ are constants and ${$ n$}$ is a positive integer.

Q: How do I use the binomial theorem to expand a binomial expression?

A: To use the binomial theorem to expand a binomial expression, you need to follow these steps:

Write the binomial expression in the form ${$ (a+b)^n$}$.
Use the formula for the binomial coefficient, ${$ \binom{n}{k} = \frac{n!}{k!(n-k)!}$}$, to calculate the coefficients of each term in the expansion.
Write out the terms of the expansion, using the calculated coefficients and the variables ${$ a$}$ and ${$ b$}$.

Q: What is Pascal's triangle?

A: Pascal's triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. The first few rows of Pascal's triangle are:

${$ \begin{array}{cccccccc} & & & 1 & & & & \ & & 1 & & 1 & & & \ & 1 & & 2 & & 1 & & \ & 1 & & 3 & & 3 & & 1 \ & 1 & & 4 & & 6 & & 4 & 1 \ & 1 & & 5 & & 10 & & 10 & 5 & 1 \ & 1 & & 6 & & 15 & & 20 & 15 & 6 & 1 \ & 1 & & 7 & & 21 & & 35 & 35 & 21 & 7 & 1 \ & 1 & & 8 & & 28 & & 56 & 70 & 56 & 28 & 8 & 1 \ & 1 & & 9 & & 36 & & 84 & 126 & 126 & 84 & 36 & 9 & 1 \ & 1 & & 10 & & 45 & & 120 & 210 & 252 & 210 & 120 & 45 & 10 & 1 \ \end{array}$}$

Q: How do I use Pascal's triangle to expand a binomial expression?

A: To use Pascal's triangle to expand a binomial expression, you need to follow these steps:

Write the binomial expression in the form ${$ (a+b)^n$}$.
Use the entries in Pascal's triangle to calculate the coefficients of each term in the expansion.
Write out the terms of the expansion, using the calculated coefficients and the variables ${$ a$}$ and ${$ b$}$.

Q: What is the difference between the binomial theorem and Pascal's triangle?

A: The binomial theorem is a mathematical formula that describes the expansion of a binomial expression, while Pascal's triangle is a triangular array of numbers that can be used to calculate the coefficients of each term in the expansion.

Q: How do I use the binomial theorem to approximate the value of a binomial expression?

A: To use the binomial theorem to approximate the value of a binomial expression, you need to follow these steps:

Write the binomial expression in the form ${$ (a+b)^n$}$.
Use the binomial theorem to expand the expression up to a certain power of ${$ x$}$.
Substitute the values of ${$ a$}$, ${$ b$}$, and ${$ n$}$ into the expanded expression.
Evaluate the expression using a calculator or by hand.

Q: What is the advantage of using the binomial theorem to approximate the value of a binomial expression?

A: The advantage of using the binomial theorem to approximate the value of a binomial expression is that it allows you to calculate the value of the expression to a high degree of accuracy, even when the expression is complex or difficult to evaluate directly.

Q: What is the disadvantage of using the binomial theorem to approximate the value of a binomial expression?

A: The disadvantage of using the binomial theorem to approximate the value of a binomial expression is that it can be time-consuming and difficult to calculate the coefficients of each term in the expansion, especially for large values of ${$ n$}$.

Q: How do I choose the right method for expanding a binomial expression?

A: To choose the right method for expanding a binomial expression, you need to consider the following factors:

The complexity of the expression: If the expression is simple, you may be able to use the binomial theorem directly. If the expression is complex, you may need to use Pascal's triangle or another method.
The accuracy required: If you need to calculate the value of the expression to a high degree of accuracy, you may need to use the binomial theorem or another method that allows for high accuracy.
The time and effort required: If you are short on time or effort, you may want to use a simpler method, such as Pascal's triangle, to expand the expression.

Q: What are some common applications of the binomial theorem?

A: The binomial theorem has many common applications in mathematics, science, and engineering, including:

Calculating the probability of independent events
Modeling population growth and decay
Calculating the area and volume of shapes
Modeling chemical reactions and physical processes

Q: What are some common mistakes to avoid when using the binomial theorem?

A: Some common mistakes to avoid when using the binomial theorem include:

Failing to calculate the coefficients of each term in the expansion
Failing to evaluate the expression correctly
Using the wrong method for expanding the expression
Failing to consider the accuracy required for the calculation

Q: How do I troubleshoot common errors when using the binomial theorem?

A: To troubleshoot common errors when using the binomial theorem, you need to follow these steps:

Check your calculations: Make sure that you have calculated the coefficients of each term in the expansion correctly.
Check your evaluation: Make sure that you have evaluated the expression correctly.
Check your method: Make sure that you are using the right method for expanding the expression.
Check your accuracy: Make sure that you have considered the accuracy required for the calculation.

By following these steps, you can troubleshoot common errors when using the binomial theorem and ensure that you get accurate results.