Question

Show that
C₀ + C₂ + C₄ + C₆ + C₈ = C₁ + C₃ + C₅ + C₇ = 128​

Answer 1

Hi,

Consider the expansion of (1 + x)⁸ = C₀ + C₁x + C₂x² + C₃x³ + C₄x⁴ + C₅x⁵

+ C₆x⁶ + C₇x⁷ + C₈x⁸-------------------(1)

Substituting x = -1 in the above expression , we get

(1 - 1)⁸ = C₀ - C₁ + C₂ - C₃ + C₄ - C₅ + C₆ - C₇ + C₈

⇒C₀ - C₁ + C₂ - C₃ + C₄ - C₅ + C₆ - C₇ + C₈ = 0

⇒C₀ + C₂ + C₄ + C₆ + C₈ = C₁ + C₃ + C₅ + C₇ -------(2)

Substituting x = 1 in (1), we get

(1 + 1)⁸ = C₀ + C₁ + C₂ + C₃ + C₄ + C₅ + C₆ + C₇ + C₈

⇒C₀ + C₁ + C₂ + C₃ + C₄ + C₅ + C₆ + C₇ + C₈ = 256

⇒(C₀ + C₂ + C₄ + C₆ + C₈) + (C₁ + C₃ + C₅ + C₇) = 256

Using (2), we get

C₀ + C₂ + C₄ + C₆ + C₈ = (C₁ + C₃ + C₅ + C₇) = 128

Hope, it helps !

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