Find the coefficient of x⁵ in the expansion of (x+1/(2x))⁹

To find the coefficient of \( x^5 \) in the expansion of \( \left(x + \frac{1}{2x}\right)^9 \), we can use the Binomial Theorem:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case:
- \( a = x \)
- \( b = \frac{1}{2x} \)
- \( n = 9 \)

So the general term in the expansion is:

\[
T_k = \binom{9}{k} \cdot x^{9-k} \cdot \left(\frac{1}{2x}\right)^k
\]

Simplifying \( T_k \):

\[
T_k = \binom{9}{k} \cdot x^{9-k} \cdot \frac{1}{(2x)^k} = \binom{9}{k} \cdot \frac{x^{9-k}}{2^k \cdot x^k} = \binom{9}{k} \cdot \frac{x^{9 - 2k}}{2^k}
\]

Now, we need the coefficient of \( x^5 \), so we set up the equation:

\[
9 - 2k = 5
\]

Solving for \( k \):

\[
2k = 4 \Rightarrow k = 2
\]

Now, substitute \( k = 2 \) back into \( T_k \) to find the coefficient:

\[
T_2 = \binom{9}{2} \cdot \frac{x^{9 - 2 \cdot 2}}{2^2} = \binom{9}{2} \cdot \frac{x^5}{4}
\]

Calculating \( \binom{9}{2} \):

\[
\binom{9}{2} = \frac{9 \cdot 8}{2 \cdot 1} = 36
\]

Thus,

\[
T_2 = 36 \cdot \frac{x^5}{4} = 9x^5
\]

So, the coefficient of \( x^5 \) in the expansion of \( \left(x + \frac{1}{2x}\right)^9 \) is **9**.

Would you like further clarification on this solution? 

---

1. What is the Binomial Theorem and how is it applied in different contexts?
2. How would the coefficient change if the power were \(10\) instead of \(9\)?
3. What role does the term \(k\) play in identifying specific terms in expansions?
4. How would you solve this if the expression was \((x + 1/x)^9\) instead?
5. Can we solve this problem by other methods like generating functions?

**Tip:** Understanding the general term in binomial expansions helps in identifying specific terms more efficiently.

Learn how to determine the coefficient of x⁵ in the binomial expansion of (x + 1/(2x))⁹ using the Binomial Theorem. This solution includes step-by-step calculations of binomial coefficients and exponent simplifications, ideal for high school algebra students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation