Question

In a certain test ${\mathrm{a}}_{\mathrm{i}}$ students gave wrong answers to at least $\mathrm{i}$ questions, where $\mathrm{i}=1,2,3...\mathrm{k}$ No student gave more than $\mathrm{k}$ wrong answers. The total number of wrong answers is

• A. ${\mathrm{a}}_1+{\mathrm{a}}_2...+{\mathrm{a}}_{\mathrm{k}}$
• B. ${\mathrm{a}}_1+2{\mathrm{a}}_2...+{\mathrm{ka}}_{\mathrm{k}}$
• C. ${\mathrm{a}}_1+{\mathrm{a}}_2...+{\mathrm{a}}_{\mathrm{k}}-\mathrm{k}$
• D. $\frac{\mathrm{k}}{2\left(\mathrm{k}+1\right)}$

Answer 1

The correct option is A

${\mathrm{a}}_1+{\mathrm{a}}_2...+{\mathrm{a}}_{\mathrm{k}}$

Explanation for the correct option:

Finding the total number of wrong answers:

Total number of students who give wrong answers$={\mathrm{a}}_{\mathrm{i}}$

Total number of students who give $\mathrm{i}+1$ wrong answers$={\mathrm{a}}_{\mathrm{i}}+1$

Total number of students who give $\mathrm{i}$ wrong answers $={\mathrm{a}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{i}+1}$

Total number of answers given by the students:

$$\begin{gathered} =1(a_1-a_2)+2(a_2-a_3)....(k-1)(a_{k-1}-a_k)-k{a}_k \\ =a_1+a_2+a_3.....+a_{k.} \end{gathered}$$

Hence the correct option is A.