Question

In certain tests, there are n questions. In the test \[{{2}^{^{i-1}}}\] students gave wrong answers to at least \[i\] questions where \[i\] = 1, 2, 3, 4. . . . . . . . . . . . . . . . . . .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

Hint: In the question it was given that \[{{2}^{^{i-1}}}\] students gave wrong answers to at least \[i\] questions. It follows the G.P (Geometric progression) because it was given at least \[i\]. So using this approach we will solve this problem.
Complete step-by-step answer:
Here the progression follows as
 \[\begin{align}
  & {{2}^{0}}={{2}^{1}}-1 \\
 & {{2}^{0}}+{{2}^{1}}={{2}^{2}}-1 \\
 & {{2}^{0}}+{{2}^{1}}+{{2}^{2}}={{2}^{3}}-1 \\
 & {{2}^{0}}+{{2}^{1}}+{{2}^{2}}+{{2}^{3}}={{2}^{4}}-1 \\
\end{align}\]
\[{{2}^{0}}+{{2}^{1}}+{{2}^{2}}+{{2}^{3}}+{{2}^{4}}={{2}^{5}}-1\] . . . . . . . . . . . . . . . . . . .. . . . . . . . .continuing further,
It was given at least \[i\] so we get the above progression.
Now 2047 can be written as 2048 - 1 and we know that \[{{2}^{11}}=2048\].
Therefore it is the sum of powers of 2 until n=11.
Therefore the answer is 11.
The answer is option B.
Note: In the problem it was given at least \[i\] wrong answers that means there might be 1 wrong answer or 2 wrong answers. . . . . . . . \[i\]. So we have to write the G.P and solve to get the total number of questions.