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If (1 + x + x 2 ) 25 = a 0 + a 1 x + a 2 x 2 + ..... + a 50 . x 50 then a 0 + a 2 + a 4 + ..... + a 50 is : 1. even 2. odd & of the form 3n 3. odd & of the form (3n - 1) 4. odd & of the form (3n + 1)
























If (1 + x + x2)25 = a0 + a1x + a2x2 + ..... + a50 . x50 then a0 + a2 + a4 + ..... + a50 is :
even
odd & of the form 3n
odd & of the form (3n - 1)
odd & of the form (3n + 1)

Grade:12th Pass

2 Answers

Nehal Wani
21 Points
12 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
24 Points
one year 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=a0a1+a2a3...+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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