Question

Show that
$\rm C_{1}+C_{2}+C_{3}+ ... +C_{n}=2^{n}-1}$

Answer 1

Answer:

C₁ + C₂ + C₃ + ............ + Cn = 2ⁿ - 1

Step-by-step explanation:

Hi,

Consider the expansion of (1 + x)ⁿ

= C₀ + C₁x + C₂x² + C₃x³ + ........ +Cn -----------------------(1)

Substituting x = 1 in (1), we get

(1 + 1)ⁿ = C₀ + C₁ + C₂ + C₃ + ............ + Cn

Re-arranging the terms, we get

⇒C₀ + C₁ + C₂ + C₃ + ............ + Cn = 2ⁿ

But, we know that C₀ = 1, hence

C₁ + C₂ + C₃ + ............ + Cn = 2ⁿ - 1

Hope, this helps !