Question

If the expansion of (1 + x)ⁿ, C₀ + C₁ + C₂ + C₃ + … C_n are coefficients different terms then find the value C₀ + C₂ + C₄… .

Answer 1

(1 + x)ⁿ= ⁿC₀ 1ⁿ + ⁿC₁ 1ⁿ + ⁿC₁ 1^n-1 x¹

ⁿC₂ 1^n-2 x² + ⁿC₃ 1^n-3x³ +….

Putting x = 1

(1 + 1)ⁿ = ⁿC₀ + ⁿC₁ +ⁿC₂ + ⁿC₃ + ….

Putting x = – 1

(1 – 1)ⁿ = ⁿC₀ – ⁿC₁ +ⁿC₂ – ⁿC₃ + ….

Here ⁿC₀ + ⁿC₁ +ⁿC₂ + ⁿC₃ + …. = 2ⁿ … (i)

“C0 – “Ci + ”C2 – “C3 + … = 0 … (ii)

Adding equation (i) and (ii)

2[ⁿC₀ + ⁿC₂ + ⁿC₄ + …] = 2ⁿ

⇒ ⁿC₀ + ⁿC₂ + ⁿC₄ + … = 2^{n – 1}

or C₀ + C₂ + C₄ + … = 2^{n – 1}