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Coefficient Conclusion (Posted on 20140808) 

Consider the polynomial
P(y) = (y+1)(y^{2} + 2)(y^{4} + 4)(y^{8} + 8)...(y^{1024} + 1024)
Determine the value of A, given that the coefficient of y^{2012} in the expansion of P(y)
is equal to 2^{A}
Solution

Comment 1 of 1

Product[r=0,…10
of (y^2^{r} + 2^r)].
Since each bracket contains y raised to a different power of 2, the
binary representation, 2012 = 11111011100_{2}, tells us that y^{2012}
derives
from the product y^(2^{2}+2^{3}+2^{4}+2^{6}+2^{7}+2^{8}+2^{9}+2^{10}),
these particular
powers coming uniquely from 8 of the 11 brackets. The coefficient of
y^{2012} will therefore come from the powers of 2 in the ‘missing’ 3
brackets.
Thus: A = coefficient of y^{2012}
= 2^(0 + 1 + 5) = 64.

Posted by Harry
on 20140810 16:53:40 


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