https://file.aiquickdraw.com/mathbot/file/1727723678193xv5x7tls.jpg

The problem is to find the sixth term of the expansion of \(\left( x - \frac{1}{x} \right)^{10}\).

To solve this problem, we use the Binomial Theorem, which is used to expand expressions of the form \((a + b)^n\). The general term in the expansion is given by:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

### Step 1: Identifying the Components
In this case:
- \(a = x\)
- \(b = -\frac{1}{x}\)
- \(n = 10\)

We need to find the sixth term, which corresponds to \(T_6\). In the binomial expansion, the \(k\)-th term is expressed as:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

For the sixth term (\(k+1 = 6\)), we have:
- \(k = 5\)

### Step 2: Apply the Formula
The sixth term is:

\[
T_6 = \binom{10}{5} x^{10-5} \left( -\frac{1}{x} \right)^5
\]

### Step 3: Simplify the Expression
First, calculate \(\binom{10}{5}\):

\[
\binom{10}{5} = \frac{10!}{5!(10-5)!} = 252
\]

Now substitute back:

\[
T_6 = 252 \cdot x^5 \cdot \left( -\frac{1}{x} \right)^5
\]

Simplify the powers of \(x\):

\[
T_6 = 252 \cdot x^5 \cdot \left( -1 \right)^5 \cdot x^{-5}
\]

\[
T_6 = 252 \cdot (-1)^5 = -252
\]

### Step 4: Conclusion
The sixth term of the expansion is \(-252\).

### Final Answer
The correct answer is:

**d) -252**

---

Would you like more details on any part of this solution or have any other questions?

### Related Questions:
1. How do we find the general term in a binomial expansion?
2. What is the Binomial Theorem and how is it applied in expansions?
3. Can you explain why the sign alternates in the binomial expansion of \((a - b)^n\)?
4. How do you determine the coefficient of a particular term in a binomial expansion?
5. What are the properties of binomial coefficients?

### Tip:
Remember that the sign in the binomial expansion depends on the power of the negative term. For \((-b)^k\), if \(k\) is odd, the term will be negative, and if \(k\) is even, it will be positive.

Hallar el sexto término del desarrollo: (x - 1/x)^10

Learn how to find the sixth term in the expansion of (x - 1/x)^10 using the Binomial Theorem. This problem involves algebraic binomial expansion and calculating binomial coefficients. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation