Write the middle term in the expansion of \((x-\frac{1}{2x})^{6}\) .

1.	Write the middle term in the expansion of \((x-\frac{1}{2x})^{6}\) .
Answer» The general term in the expansion of \((x-\frac{1}{2x})^6\) is T_{r + 1} = (– 1)^r ⁶C_r x^{6 – r} \((\frac{1}{2x})^r\) = (– 1)^r . ⁶C_r x^{6 – r}. 2^{– r} x^{– r} = (– 1)^r . ⁶C_r x^{6 – 2r} . 2^{– r} Now the power of binomial expansion being 6, (even), the middle term is \((\frac{6}{2}+1)th\) term = 4th term. ∴ T₄ = T₃₊₁ = (-1)³ . ⁶C₃ x^6-6 2^-3 = \((-1) \times \frac{6\times 5 \times 4}{3 \times 2 \times 1} \times x^0 \times \frac{1}{8} = -\frac{5}{2}\)

Answer»

The general term in the expansion of \((x-\frac{1}{2x})^6\) is

T_{r + 1} = (– 1)^r ⁶C_r x^{6 – r} \((\frac{1}{2x})^r\) = (– 1)^r . ⁶C_r x^{6 – r}. 2^{– r} x^{– r}

= (– 1)^r . ⁶C_r x^{6 – 2r} . 2^{– r}

Now the power of binomial expansion being 6, (even), the middle term is \((\frac{6}{2}+1)th\) term = 4th term.

∴ T₄ = T₃₊₁ = (-1)³ . ⁶C₃ x^6-6 2^-3

= \((-1) \times \frac{6\times 5 \times 4}{3 \times 2 \times 1} \times x^0 \times \frac{1}{8} = -\frac{5}{2}\)

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