If the expansion of (1 + x)n, C0 + C1 + C2 + C3 + … Cn are co

1.	If the expansion of (1 + x)n, C0 + C1 + C2 + C3 + … Cn are coefficients different terms then find the value C0 + C2 + C4… .
Answer» (1 + x)ⁿ= ⁿC₀1ⁿ + ⁿC₁1ⁿ+ ⁿC₁1^n-1x¹ ⁿC₂1^n-2x² + ⁿC₃1^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}

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

Answer»

(1 + x)ⁿ= ⁿC₀1ⁿ + ⁿC₁1ⁿ+ ⁿC₁1^n-1x¹

ⁿC₂1^n-2x² + ⁿC₃1^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}