binomial expansion

Question

The middle term in the expansion of (x + 1/x)¹⁰ is

RankGuru · Accepted Answer

1. **Determine Middle Term Position**: For $(a+b)^n$:
   - If $n$ is even, there is one middle term at position $(\frac{n}{2} + 1)$.
   - Here $n=10$ (even), so the middle term is the $(\frac{10}{2} + 1) = 6^{th}$ term ($r=5$).
2. **Apply General Term**: $T_6 = \binom{10}{5} (x)^{10-5} (\frac{1}{x})^5$
3. **Simplification**:
   - $T_6 = \binom{10}{5} x^5 \cdot \frac{1}{x^5} = \binom{10}{5}$.
   - Value $= \frac{10 	imes 9 	imes 8 	imes 7 	imes 6}{120} = 252$.

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