(Q). In a certain test, aᵢ students gave wrong answer to at least i question where i = 1, 2, 3, ………, k & no student gave more than k wrong answers. What is the total number of wrong answers?
Solution:-
Number of students who gave i or more than i wrong answers = aᵢ
Number of students who gave (i+1) or more than (i+1) wrong answers = aᵢ₊₁
∴ Number of students who gave wrong answers to exactly i questions = aᵢ – aᵢ₊₁
∴ total number of wrong answers =
= 1.(a₁ – a₂) + 2(a₂ – a₃) + 3.(a₃ – a₄) + ………… + (k – 1)(aₖ₋₁ – aₖ) + k(aₖ – aₖ₊₁) …………. (i)
Here aₖ₊₁ = Number of students who gave wrong answer to (k+1) or more question
& as per question no student gave more than k wrong answer
∴ aₖ₊₁ = 0
Hence equation (i) can again be written as:-
⟹ a₁ + a₂ + a₃ + ………… + aₖ
∴ total number of wrong answers
= a₁ + a₂ + a₃ + ………… + aₖ Answer