In a certain test ai students gave wrong answers to at least $i$ questions where $i = 1, 2, 3, . . . . k$ .
No student gave more than $k$ wrong answers .
The total numbers of wrong answers given is?

$a_{1} + 2 a_{2} + 3 a_{3} + \dots . + k a_{k}$

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$a_{1} + a_{2} + a_{3} + \dots . + a_{k}$

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zero

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None of the above

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Solution

The correct option is B
$a_{1} + a_{2} + a_{3} + \dots . + a_{k}$

Explanation for correct answer:
Students who gave answer to only $1$ question $= a_{1} - a_{2}$
Now the students who gave answer to 2 questions $= a_{2} - a_{3}$
Similarly, for students who gave answer to $k$ questions $= a_{k - 1} - a_{k}$
So, total number of wrong answers,
$\Rightarrow 1 (a_{1} - a_{2}) + 2 (a_{2} - a_{3}) + 3 (a_{3} - a_{4}) + \dots + k (a_{k - 1} - a_{k})$
On further simplifying above equation, we get,
$\Rightarrow a_{1} - a_{2} + 2 a_{2} - 2 a_{3} + 3 a_{3} - 3 a_{4} + \dots + k a_{k - 1} - k a_{k} = a_{1} + a_{2} + a_{3} + \dots + a_{k} $
Hence, the correct option is B.

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