Question

The coefficient of $x^{15}$ in the product of $(1-x)(1-2x)(1-2^{2}x)....(1-2^{15}x)$ is equal to

• A. $2^{105}-2^{121}$
• B. $2^{121}-2^{105}$
• C. $2^{120}-2^{104}$
• D. None of the above.

Answer 1

The correct option is A $2^{105}-2^{121}$

$(1-2^{0}x)(1-2^{1}x)(1-2^{2}x).....(1-2^{15}x)$

Maximum power of x$=16$

Power of $x^{15}=1+2+3+...+15=120$

We can tell the coefficient of $x^{15}$ $=(-1)(2^{120}+2^{120-1}+2^{120-2}+....+2^{120-15}$

$=(-1)\left(2^{120}\right)(1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{15}})\Downarrow$

This is in G.P. whose sum$=\frac{1-{\left(\frac{1}{2}\right)}^{16}}{\frac{1}{2}}$

$=(-1)\left(2^{120}\right)\times 2\times (1-{\left(\frac{1}{2}\right)}^{16})$

$=2^{105}-2^{121}$