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

Answer

To solve the problem \( (1)(2020)+(2)(2019)+(3)(2018)+\ldots+(2020)(1)=2020 \times 2021 \times k \), we will follow these steps:

### Step 1: Rewrite the Series
The series can be rewritten in summation notation:
\[ S = \sum_{r=1}^{2020} r \cdot (2021 - r) \]
This represents the sum of products where \( r \) varies from \( 1 \) to \( 2020 \).

### Step 2: Expand the Summation
Expanding the summation gives:
\[ S = \sum_{r=1}^{2020} (2021r - r^2) = 2021 \sum_{r=1}^{2020} r - \sum_{r=1}^{2020} r^2 \]

### Step 3: Use Summation Formulas
We will use the formulas for the sum of the first \( n \) natural numbers and the sum 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 \( n = 2020 \):
\[ \sum_{r=1}^{2020} r = \frac{2020 \times 2021}{2} \]
\[ \sum_{r=1}^{2020} r^2 = \frac{2020 \times 2021 \times 4041}{6} \]

### Step 4: Substitute Back into the Summation
Substituting these results back into our expression for \( S \):
\[ S = 2021 \left(\frac{2020 \times 2021}{2}\right) - \left(\frac{2020 \times 2021 \times 4041}{6}\right) \]

### Step 5: Factor Out Common Terms
Factoring out \( 2020 \times 2021 \):
\[ S = 2020 \times 2021 \left(\frac{2021}{2} - \frac{4041}{6}\right) \]

### Step 6: Simplify the Expression
Finding a common denominator (which is 6):
\[ \frac{2021}{2} = \frac{6063}{6} \]
Thus,
\[ S = 2020 \times 2021 \left(\frac{6063 - 4041}{6}\right) = 2020 \times 2021 \left(\frac{2022}{6}\right) \]
This simplifies to:
\[ S = 2020 \times 2021 \times 337 \]

### Step 7: Compare with Given Expression
We know from the problem statement that:
\[ S = 2020 \times 2021 \times k \]
Thus, we can equate:
\[ k = 337 \]

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

### Final Answer
\[ \frac{k}{100} = 3.37 \]