If `(1)(2020)+(2)(2019)+(3)(2018)+…….+(2020)(1)=2020xx2021xxk,` then the value of `(k)/(100)` is equal to

Answer

To solve the problem, we need to evaluate the sum \( S = (1)(2020) + (2)(2019) + (3)(2018) + \ldots + (2020)(1) \) and express it in the form \( 2020 \times 2021 \times k \).

### Step 1: Express the sum in a simplified form
The sum can be expressed as:
\[ S = \sum_{r=1}^{2020} r \cdot (2021 - r) \]
This is because the first term corresponds to \( r=1 \) and \( 2021 - r \) gives us the decreasing sequence from 2020 to 1.

### Step 2: Expand the summation
Now we can expand the summation:
\[ S = \sum_{r=1}^{2020} (2021r - r^2) \]
This can be split into two separate summations:
\[ S = 2021 \sum_{r=1}^{2020} r - \sum_{r=1}^{2020} r^2 \]

### Step 3: Use summation formulas
We can use the formulas for the summation of the first \( n \) natural numbers and the summation of the squares of the first \( n \) natural numbers:
- The sum of the first \( n \) natural numbers:
\[ \sum_{r=1}^{n} r = \frac{n(n+1)}{2} \]
- The sum of the squares of the first \( n \) natural numbers:
\[ \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6} \]

For \( n = 2020 \):
\[ \sum_{r=1}^{2020} r = \frac{2020 \cdot 2021}{2} \]
\[ \sum_{r=1}^{2020} r^2 = \frac{2020 \cdot 2021 \cdot 4041}{6} \]

### Step 4: Substitute the values into the equation
Substituting these values back into our expression for \( S \):
\[ S = 2021 \cdot \frac{2020 \cdot 2021}{2} - \frac{2020 \cdot 2021 \cdot 4041}{6} \]

### Step 5: Factor out common terms
Factoring out \( 2020 \cdot 2021 \):
\[ S = 2020 \cdot 2021 \left( \frac{2021}{2} - \frac{4041}{6} \right) \]

### Step 6: Simplify the expression inside the parentheses
To simplify \( \frac{2021}{2} - \frac{4041}{6} \), we need a common denominator:
\[ \frac{2021 \cdot 3}{6} - \frac{4041}{6} = \frac{6063 - 4041}{6} = \frac{2022}{6} = \frac{337}{1} \]

### Step 7: Final expression for \( S \)
Thus, we have:
\[ S = 2020 \cdot 2021 \cdot 337 \]
This means \( k = 337 \).

### Step 8: Find \( \frac{k}{100} \)
Finally, we need to find \( \frac{k}{100} \):
\[ \frac{k}{100} = \frac{337}{100} = 3.37 \]

### Final Answer
Therefore, the value of \( \frac{k}{100} \) is:
\[ \boxed{3.37} \]