Find the coeff. of x15 in the expansion of (1 - x)(1 - 2x)(1 - 22x).............(1 - 215x). Please help me its urgent.
1057the answer is 216-1 bcoz in a polynomial where there are 16 terms the co-efficient of x15 is the sum of all the co-efficients of x.....
in this case it is 20 + 21 + 22 + 23.... + 215 which is a GP wid first term 1 nd common ratio 2 and the no of terms 16.....
1i'm getting -2121.
totally we have 16 x's so for getting x15 we have to select x 15 times or in other words in each time neglect 1 'x'.
1057@rishabh observe that in a quadratic and cubic equation also co- efficient of xn-1 is the sum of the roots....
it will be the same here...
1agree with the person in the second post. answer should be 216 - 1.
1057when u only take de first 2 terms the co-efficient of x is -3 which is -(20 + 21)
when u take de first three terms then the co-efficient of x2 is -7(20 + 21 + 22)
therefore when there are 16 terms the co-efficient of x15 shud be -(216 - 1)
simple observation i guess..
3see i did this.
(1-x)(1-2x) = 1 + 2x^2 - 3x.
coefficient of x is -3.
-3 = -2 - 1.
(1-x)(1-2x)(1-4x) = -8 x^3+14 x^2-7 x+1
coefficient of x^2 is 14.
14 = (-4)(-2) + (-4)(-1) + (-2)(-1).
(1-x)(1-2x)(1-4x)(1-8x) = 64 x^4-120 x^3+70 x^2-15 x+1
coefficient of x^3 is -120.
-120 = (-8)(-4)(-2) + (-8)(-2)(-1) + (-4)(-2)(-1) + (-1)(-8)(-4).
now i have a doubt.
to get x^15 we have to multiply 15 terms out of the given 16 terms.
so we have to multiply any 15 and leave one term.
i have illustrated the coefficient of x^3 in (1-x)(1-2x)(1-4x)(1-8x) above.
what is wrong with my observation???
3i just put this on wolfram alpha.
the coefficient of x^15 is
-2658415426750624442466766226058117120 ...
visit this link.
http://www.wolframalpha.com/input/?i=n+%3D+0+to+n+%3D+15+product+of+(1-(2^n)x)
and moreover 2^16 - 1 is just {65535}..
isn't that a big difference????
1057oh sorry i did a mistake...
co-efficient of x will be -(2^16-1)
co-efficient of x^15 will be product of co-efficient of x's taking 15 at a time leaving 1 of the x's co-efficient........
1057the answer shud be 2120+2119+2118+2117+........2105
2105(216-1) is the answer....
1057@hemang co-efficient of x14 can also be found out if we take the co-efficient of x 14 at a time but it is a tedious job as the no of terms in the desired sum will be 16C2 which is 120 nd the exam time will be over by the time u find the answer.... :P
nd u are correct it will be -2105(216-1)
