In a certain test, a 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

Answer

To solve the problem, we need to calculate the total number of wrong answers given by students in a test, where each student has given a varying number of wrong answers ranging from at least 1 to at most k.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( a_i \) be the number of students who gave exactly \( i \) wrong answers. Here, \( i \) can take values from 1 to \( k \).

2. **Calculate the Number of Students**:
- The number of students who gave exactly 1 wrong answer is \( a_1 \).
- The number of students who gave exactly 2 wrong answers is \( a_2 - a_3 \) (students who gave at least 2 wrong answers minus those who gave more than 2).
- Similarly, for 3 wrong answers, it is \( a_3 - a_4 \), and so on.
- For \( k \) wrong answers, it is \( a_{k-1} - a_k \).

3. **Total Wrong Answers Calculation**:
The total number of wrong answers can be calculated as follows:
\[ \text{Total Wrong Answers} = 1 \cdot a_1 + 2 \cdot (a_2 - a_3) + 3 \cdot (a_3 - a_4) + \ldots + k \cdot (a_{k-1} - a_k) \]

4. **Simplifying the Expression**:
- This can be rewritten in a more compact form:
\[ \text{Total Wrong Answers} = a_1 + (2a_2 - 2a_3) + (3a_3 - 3a_4) + \ldots + (k a_{k-1} - k a_k) \]
- By combining like terms, we can express the total number of wrong answers as:
\[ \text{Total Wrong Answers} = a_1 + a_2 + a_3 + \ldots + a_k \]

5. **Final Result**:
Thus, the total number of wrong answers given by all students is:
\[ \text{Total Wrong Answers} = a_1 + a_2 + a_3 + \ldots + a_k \]