In a certain test, $a_{i}$ students gave wrong answers to at least i questions, where i = 1, 2, ......, k. No student gave more than k wrong answers. The total number of wrong answers given is ___.

a_{1} + a_{k}

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\frac{a k^{2}}{2}

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a_{1} - a_{2} + a_{3} - . . . . . . . . . + (- 1)^{k + 1} a_{k}

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a_{1} + a_{2} + a_{3} + . . . . . . + a_{k}

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Solution

The correct option is D $a_{1} + a_{2} + a_{3} + . . . . . . + a_{k}$
The number of students who gave wrong answers to exactly one question is $a_{1} - a_{2}$ .
The number of students who gave wrong answers to exactly two questions is $a_{2} - a_{3}$ .
The number of students who gave wrong answers to exactly three questions is $a_{3} - a_{4}$ .
The number of students who gave wrong answers to exactly k-1 questions is $a_{k - 1} - a_{k}$ .
Therefore, total number of wrong answers is
$1 (a_{1} - a_{2}) + 2 (a_{2} - a_{3}) + 3 (a_{3} - a_{4}) + . . . . . . + (k - 1) (a_{k - 1} - a_{k}) + k (a_{k}) = a_{1} + a_{2} + a_{3} + . . . . + a_{k}$

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