If (1 + x)n = a0 + a1x + a2x2 + a3x3 +...........+ anxn, prove that 2n = (a0 - a2 + a4-......) + (a1 -
a3 + a5-......).
Answers
Given : (1 + x)ⁿ = a₀ + a₁x + a₂x² + a₃x³ + ...........................+ aₙxⁿ
To Find : Prove that 2ⁿ = ( a₀ - a₂ + a₄ - ........)² + (a₁ - a₃ + a₅ - ................)²
Solution:
(1 + x)ⁿ = a₀ + a₁x + a₂x² + a₃x³ + ...........................+ aₙxⁿ
x = i
=> ( 1 + i)ⁿ = a₀ + a₁(i) + a₂(i)² + a₃(i)³ + ...........................+ aₙ(i)ⁿ
=> ( 1 + i)ⁿ = a₀ + a₁i - a₂ - a₃i + ...........................+ aₙ
=> ( 1 + i)ⁿ = ( a₀ - a₂ + a₄ - ........) + i (a₁ - a₃ + a₅ - ................)
1 + i = √2 ( 1/ √2 + i/ √2 )
=> 1 + i =√2 ( Cos(π/4) + iSinπ/4)
( 1 + i)ⁿ = {√2 ( Cos(π/4) + iSinπ/4) }ⁿ
=> ( 1 + i)ⁿ = (√2)ⁿ ( Cos(nπ/4) + iSin(nπ/4)
Equating imaginary & real part
( a₀ - a₂ + a₄ - ........) = (√2)ⁿ ( Cos(nπ/4) )
(a₁ - a₃ + a₅ - ................) = (√2)ⁿ ( Sin(nπ/4) )
( a₀ - a₂ + a₄ - ........)² + (a₁ - a₃ + a₅ - ................)² =( (√2)ⁿ)² (Cos²(nπ/4)) + ( (√2)ⁿ)² (Sin²(nπ/4))
= 2ⁿ ( Cos²(nπ/4) + Sin²(nπ/4))
= 2ⁿ ( 1)
= 2ⁿ
QED
Hence Proved
2ⁿ = ( a₀ - a₂ + a₄ - ........)² + (a₁ - a₃ + a₅ - ................)²
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