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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)

VISHWESH RAVI , 15 Years ago
Grade 12th Pass
anser 2 Answers
Nehal Wani

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!

 

Last Activity: 15 Years ago
Ankana Das

(1+x+x2)25=a0​+a1​x+a2​x2+...a50​x50


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.
Last Activity: 4 Years ago
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