Question

If $x$ is a rational number satisfying
$(1-x)(1+x+x^2+x^3+x^4)$$=\frac{31}{32}.$
Then $1+x+x^2+x^3+x^4+x^5$ is

• A. $\frac{31}{64}$
• B. $\frac{31}{32}$
• C. $\frac{63}{64}$
• D. $\frac{63}{32}$

Answer 1

The correct option is D

$\frac{63}{32}$

$(1-x)(1+x+x^2+x^3+x^4)=1-x^5=\frac{31}{32}=1-\frac{1}{32}{x}^5=\frac{1}{32}=\frac{1}{2^5}\text{or}x=\frac{1}{2}\therefore 1+x+x^2+x^3+x^4+x^5=\frac{1-x^6}{1-x}=\frac{1-{\left(\frac{1}{2}\right)}^6}{1-\frac{1}{2}}=\frac{63}{64}\times 2=\frac{63}{32}$