Let [x] denote greatest integer less than or equal to x. If for n ∈ N, (1 - x + x²)? = ∑(j=0 to 3n) a?x?, then ∑(j=0 to [3n/2]) a?? + 4∑(j=0 to [(3n-1)/2]) a???? is equal to :
Let [x] denote greatest integer less than or equal to x. If for n ∈ N, (1 - x + x²)? = ∑(j=0 to 3n) a?x?, then ∑(j=0 to [3n/2]) a?? + 4∑(j=0 to [(3n-1)/2]) a???? is equal to :
Option 1 -
1
Option 2 -
n
Option 3 -
2ⁿ⁻¹
Option 4 -
2
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1 Answer
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Correct Option - 1
Detailed Solution:(1 - x + x²)³? = ∑ a? x? (from j=0 to 3n)
= a? + a? x + a? x² + . + a? x³? (I)
Let A = a? + a? + a? + .
Let B = a? + a? + a? + .
In (I) put x = 1: (1 - 1 + 1)³? = 1.
1 = a? + a? + a? + a? + . (A + B = 1)
In (I) put x = -1: (1 - (-1) + (-1)²)³? = 3³?
3³? = a? - a? + a? - a? + . (A - B = 3³? )
(This seems incorrect based on the provided solution. Following the image:)
In (I) put x = -1, (1+1+1)^n = 1. (There must be a typo in the original problem, probably (1-x+x²)^n).
Assuming (1-x+x²)^n. Put x=-1 gives 3^n.
The provided text says putting x=-1 gives 1.
1 = a? - a? + a? - a? + .
Addin...more
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