In a certain test there are $n$ questions, in this test $2^{n-i}$ students gave wrong answer to at least $i$ questions where $i=1,2,3,\ldots,n$. If the total number of wrong answers given is $2047$,what is the value of $n$?
My attempt:From the question $2^{n-1}$ students gave wrong answer to at least one question, $2^{n-2}$ students to at least 2 and so on.Hence there's one student who answered all $n$ questions wrong.Now $(2^{n-1})-(2^{n-2})$ gives number of wrong questions by those students who got one question wrong. Continuing this way I arrived at $(2^{n-1})-1=2047$ which gave me $n=11$. Is my answer right?
Best Answer
If you add up all the powers of two from $2^{n-1}$ down to $2^0$, you'll get the total number of incorrect answers. That is just $2^n-1$. If we set that to $2047$, we get $2^n = 2048$, or $n = 11$, so you are indeed correct.
ETA: Although, you wrote $2^{n-1}-1 = 2047$, which is not right. It's $2^n-1 = 2047$.