which term is the middle term in the expansion of (a/b+b/a)^6?

The given expression is \((\frac{a}{b} + \frac{b}{a})^6\).

To find the middle term in the binomial expansion of \((x + y)^n\), the general formula for the \(r\)-th term is:

\[
T_{r+1} = \binom{n}{r} \cdot x^{n-r} \cdot y^r
\]

Here, \(n = 6\), \(x = \frac{a}{b}\), and \(y = \frac{b}{a}\).

Since \(n = 6\), the total number of terms is \(n + 1 = 7\). The middle term is the 4th term, which corresponds to \(r = 3\).

Substituting into the formula for the general term:

\[
T_4 = \binom{6}{3} \cdot \left( \frac{a}{b} \right)^{6-3} \cdot \left( \frac{b}{a} \right)^3
\]

Now, simplify the powers:

\[
T_4 = \binom{6}{3} \cdot \left( \frac{a}{b} \right)^3 \cdot \left( \frac{b}{a} \right)^3
\]

Since \(\left( \frac{a}{b} \right)^3 \cdot \left( \frac{b}{a} \right)^3 = 1\), the term simplifies to:

\[
T_4 = \binom{6}{3} \cdot 1
\]

Now, calculate \(\binom{6}{3}\):

\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\]

Thus, the middle term is:

\[
T_4 = 20
\]

### Final Answer:
The middle term in the expansion of \((\frac{a}{b} + \frac{b}{a})^6\) is **20**.

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Do you want more details or have any questions?

Here are some related questions to explore further:
1. What is the general term for the binomial expansion of \((a+b)^n\)?
2. How can we find the sum of all the coefficients in a binomial expansion?
3. What would be the middle term in \((x + y)^{10}\)?
4. How does Pascal’s Triangle relate to binomial expansions?
5. How do you find the coefficient of a specific term in binomial expansions?

**Tip:** Remember, the middle term in a binomial expansion occurs at \(\frac{n+1}{2}\)-th position when \(n\) is even.

Which term is the middle term in the expansion of (a/b + b/a)^6?

Explore how to find the middle term in the binomial expansion of (a/b + b/a)^6 using the binomial theorem and rational expressions. The step-by-step solution simplifies the expression and calculates the binomial coefficient to arrive at the middle term, which is 20. Ideal for students in Grades 9-12.

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