if `sum_1^20 r^3 = a ,sum_1^20 r^2 = b` then sum of products of 1,2,3,4,....20 taking two at a time is :

Options

A. `(a-b)/(2)`
B. `(a ^(2) - b^(2))/(2 )`
C. `a-b`
D. `a ^(2) - b^(2)`

Answer

To solve the problem, we need to find the sum of the products of the numbers from 1 to 20 taken two at a time. Given the sums \( \sum_{r=1}^{20} r^3 = a \) and \( \sum_{r=1}^{20} r^2 = b \), we can use the following steps:

### Step 1: Understand the Sum of Products
The sum of the products of the numbers from 1 to 20 taken two at a time can be represented mathematically as:
\[ \sum_{1 \leq i < j \leq 20} ij \]
This can be rewritten using the identity:
\[ \sum_{1 \leq i < j \leq n} ij = \frac{1}{2} \left( \left( \sum_{r=1}^{n} r \right)^2 - \sum_{r=1}^{n} r^2 \right) \]
For \( n = 20 \), we will use this identity.

### Step 2: Calculate \( \sum_{r=1}^{20} r \)
The sum of the first \( n \) natural numbers is given by the formula:
\[ \sum_{r=1}^{n} r = \frac{n(n+1)}{2} \]
Substituting \( n = 20 \):
\[ \sum_{r=1}^{20} r = \frac{20 \times 21}{2} = 210 \]

### Step 3: Use the Formula for the Sum of Products
Now, substituting into our identity:
\[ \sum_{1 \leq i < j \leq 20} ij = \frac{1}{2} \left( (210)^2 - b \right) \]
Calculating \( (210)^2 \):
\[ (210)^2 = 44100 \]
Thus, we have:
\[ \sum_{1 \leq i < j \leq 20} ij = \frac{1}{2} (44100 - b) \]

### Step 4: Substitute Values of \( a \) and \( b \)
From the problem, we know:
\[ \sum_{r=1}^{20} r^3 = a \quad \text{and} \quad \sum_{r=1}^{20} r^2 = b \]
Using the identity for cubes:
\[ \sum_{r=1}^{n} r^3 = \left( \sum_{r=1}^{n} r \right)^2 \]
Thus:
\[ a = (210)^2 = 44100 \]
Now, substituting \( a \) and \( b \) into our equation:
\[ \sum_{1 \leq i < j \leq 20} ij = \frac{1}{2} (44100 - b) \]

### Step 5: Final Calculation
We need to express the sum of products in terms of \( a \) and \( b \):
\[ \sum_{1 \leq i < j \leq 20} ij = \frac{1}{2} (a - b) \]
Substituting \( a = 44100 \):
\[ \sum_{1 \leq i < j \leq 20} ij = \frac{1}{2} (44100 - b) \]

### Conclusion
Thus, the sum of the products of the numbers from 1 to 20 taken two at a time is:
\[ \frac{1}{2} (44100 - b) \]