Question

In a certain test, there are n questions. In the test $2^{i-1}$ students gave wrong answers to at least i questions where $i=1,\mathrm{2.....}n.$ If the total number of wrong answers given is $2047,$ then n is equal to

• A. $10$
• B. $11$
• C. $12$
• D. $13$

Answer 1

The correct option is B $11$

Note: $2^0=2^1-1$

$2^0+2^1=2^2-1$

$2^0+2^1+2^2=2^3-1$

Or the sum of powers of $2$ equals $1$ less than the next power,

$\because 2048=2^{11}$

and $2047$ is one less than $2^{11}\therefore$ it is the sum of powers of $2$ till $n=11.$

Hence, the answer is $11.$