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Coefficient Conclusion (Posted on 2014-08-08) Difficulty: 3 of 5
Consider the polynomial

P(y) = (y+1)(y2 + 2)(y4 + 4)(y8 + 8)...(y1024 + 1024)

Determine the value of A, given that the coefficient of y2012 in the expansion of P(y) is equal to 2A

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1
Product[r=0,…10  of  (y^2r + 2^r)].

Since each bracket contains y raised to a different power of 2, the

binary representation, 2012 = 111110111002, tells us that y2012 derives

from the product y^(22+23+24+26+27+28+29+210), these particular

powers coming uniquely from 8 of the 11 brackets. The coefficient of

y2012 will therefore come from the powers of 2 in the ‘missing’ 3 brackets.

Thus:    A = coefficient of y2012 = 2^(0 + 1 + 5)  =  64.

  Posted by Harry on 2014-08-10 16:53:40
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