Question

The coefficient of $x^{49}$ in the expansion of $(x-1)(x-\frac{1}{2})(x-\frac{1}{2^2})\dots (x-\frac{1}{2^{49}})$ is equal to ?

• A. $+ve$ coefficient of $x$
• B. $-2(1-\frac{1}{2^{50}})$
• C. $(1-\frac{1}{2^{50}})$
• D. $-2(1-\frac{1}{2^{49}})$

Answer 1

Answer: (B)

$-2(1-\frac{1}{2^{50}})$

$(x+1)(x-\frac{1}{2})(x-\frac{1}{2^2})\dots ..(x-\frac{1}{2^{49}})$

Degree $=50$

Coefficient of $x^{49}$

$=-1-\frac{1}{2}-\frac{1}{2^2}\dots ..-\frac{1}{2^{49}}$

$=-(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{49}})$

$=-\left(\frac{1(1-\left(1/2{)}^{50}\right)}{1-\frac{1}{2}}\right)$

$=-\left(\frac{1-\frac{1}{2^{50}}}{\frac{1}{2}}\right)=-2(1-\frac{1}{2^{50}})$.