(Q). In a certain test there are n questions. In this test 2ᵏ students gave wrong answers to at least n-k questions. k = 0, 1, 2, ……n. If the number of wrong answers is 8191 then what is the value of n?
Solution:-
Number of students who gave wrong answers to at lest r questions = 2ⁿ⁻ʳ
& Number of students who gave wrong answers to at least (r+1) questions = 2ⁿ⁻⁽ʳ⁺¹⁾
∴ Number of students who gave wrong answer to exactly r questions = 2ⁿ⁻ʳ – 2ⁿ⁻⁽ʳ⁺¹⁾
Then the total number of wrong answers is
= 1.(2ⁿ⁻¹ – 2ⁿ⁻²) + 2.(2ⁿ⁻² – 2ⁿ⁻³)+ 3.(2ⁿ⁻³ – 2ⁿ⁻⁴) + ………….. + r.(2ⁿ⁻ʳ – 2ⁿ⁻⁽ʳ⁺¹⁾) + …….. + n.(2⁰)
= 2ⁿ⁻¹ + 2ⁿ⁻² + 2ⁿ⁻³ + 2ⁿ⁻⁴ + …………….. + 2⁰
this above equation is sum of a G.P. which is equal to = 2ⁿ – 1
Hence 2ⁿ – 1 = 8191
2ⁿ = 8192
2ⁿ = 2¹³
∴ n = 13 Answer