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 in .......

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

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

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

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none of the above

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Solution

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

$= a_{1} - a_{2} + 2 a_{2} - 2 a_{3} + 3 a_{3} - 3 a_{4} + . . . + k a_{k - 1} - k a_{k}$
Total number of wrong answers $= a_{1} + a_{2} + a_{3} + . . . + a_{k}$

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