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Grade 12th PassAlgebra

If (1+x+x²)²⁵ = a₀+ a₁x + a₂x² +.....+ a₅₀.x⁵⁰ then a₀ + a₂ + a₄ + ..... + a₅₀ is :

1. even

2. odd & of the form 3n

3. odd & of the form (3n-1)

4. odd & of the form (3n+1)

VISHWESH RAVI

16 Years agoGrade 12th Pass

2 Answers

Nehal Wani

16 Years ago

If (1+x+x²)²⁵ = a₀+ a₁x + a₂x² +.....+ a₅₀.x⁵⁰ then a₀ + a₂ + a₄ + ..... + a₅₀ is :

Put x=1,

a₀ + a₂ + a₄ + ..... + a₅₀=3²⁵⁼3.3²⁴

Hence (2) is the correct option!

Ankana Das

5 Years ago

(1+x+x2)25=a0+a1x+a2x2+...a50x50

Substituting x=1, we get

325=a0+a1+a2+...a50 ...(i)

Substituting x=−1 we get

1=a0−a1+a2−a3...+a50 ...(ii)

Adding i and ii, we get

325+1=2[a0+a2+a4+...a50]

Hence

a0+a2+a4+...a50=2325+1

Since the LHS is a positive integer hence the RHS has to be a positive integer.
Therefore

325+1 will be divisible by 2.

Hence

325+1 will be even.

And

32n+1+1 is always even and is divisible by 4.

Hence

2325+1 will be even.

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