Question

Constant term in expansion of \( \left(1+\frac{x}{2}-\frac{2}{x}\right)^{4} \) is

Answer 1

[1 + (x/2) – (2/x)]⁴ 

= [1 + {(x/2) – (2/x)}]⁴ 

= ⁴c₀ + ⁴c₁ [(x/2) – (2/x)] + ⁴c₂ [(x/2) – (2/x)]² + ⁴c₃ [(x/2) – (2/x)]³(2/x)]³ + ⁴c₄ [(x/2) – (2/x)]⁴ 

= 1 + 4[(x² – 4) / (2x)] + 6[(x⁴ – 8x² + 16) / (4x²)]  + [4 / (8x³)] (x⁶ – 12x⁴ + 48x² – 64) + (x⁴ / 16) – x² + 6 – (16 / x²) +  (16 / x⁴)

= 1 + 2x – (8/x) + (3/2)x² – 12 + (24 / x²) + (x³ / 2) – 6x + (24 / x) – (32 / x³) + (x⁴ / 16) – x² + 6 – (16 / x²) +  (16 / x⁴)

= – 5 – 4x + (16 / x) + (x² / 2) + (x³ / 2) + (x⁴ / 16) + (8 / x²) – (32/ x³) +  (16 / x⁴)

Hence constant term is – 5.