Question

if (1+x+2x^2)^20 =a0+a1x+a2x^2+...+a40x^40, then a0+a2+a4+...a38 is equal to

Answer 1

Answer:

(3²⁰ - 1)/2

Step-by-step explanation:

Hi,

Given (1 + x + x²)²⁰ = a₀ + a₁x + a₂x²+.......+a₄₀x⁴⁰

Substituting x = 1 we get

(1+1+1)²⁰ = a₀ + a₁ + a₂ + .......+ a₄₀

=> a₀ + a₁ + a₂ + .......+ a₄₀ = 3²⁰-----------(1)

On substituting x = -1, we get

(1 - 1 + 1)²⁰ = a₀ - a₁ + a₂.........+ a₄₀

=> a₀ - a₁ + a₂.........+ a₄₀ = 1---------(2)

Adding (1) and (2), we get

2(a₀ + a₁ + a₂.........+ a₄₀) = 3²⁰ + 1

=> a₀ + a₁ + a₂.........+ a₃₈ +  a₄₀ = (3²⁰ + 1)/2

But x⁴⁰ is the last term in the expansion and its coefficient will be 20C20 which is equal to 1

Hence,

a₀ + a₁ + a₂.........+ a₃₈ = (3²⁰ + 1)/2 - 1 = (3²⁰ - 1)/2

Hope, it helped !