If `S_(1), S_(2) and S_(3)` are the sums of first n natureal numbers, their squares and their cubes respectively, then `(S_(1)^(4)S_(2)^(2)-S_(2)^(2)S_(3)^(2))/(S_(1)^(2) +S_(2)^(2))=`

Options

A. 4
B. 2
C. 1
D. 0

Answer

To solve the problem, we need to find the value of the expression

\[ \frac{S_1^4 S_2^2 - S_2^2 S_3^2}{S_1^2 + S_2^2} \]

where \( S_1 \), \( S_2 \), and \( S_3 \) are defined as follows:

1. \( S_1 \) is the sum of the first \( n \) natural numbers.
2. \( S_2 \) is the sum of the squares of the first \( n \) natural numbers.
3. \( S_3 \) is the sum of the cubes of the first \( n \) natural numbers.

### Step 1: Calculate \( S_1 \), \( S_2 \), and \( S_3 \)

- **Sum of the first \( n \) natural numbers**:
\[ S_1 = \frac{n(n + 1)}{2} \]

- **Sum of the squares of the first \( n \) natural numbers**:
\[ S_2 = \frac{n(n + 1)(2n + 1)}{6} \]

- **Sum of the cubes of the first \( n \) natural numbers**:
\[ S_3 = \left( \frac{n(n + 1)}{2} \right)^2 \]

### Step 2: Substitute \( S_1 \), \( S_2 \), and \( S_3 \) into the expression

Now we substitute these values into the expression:

\[ \frac{S_1^4 S_2^2 - S_2^2 S_3^2}{S_1^2 + S_2^2} \]

### Step 3: Calculate \( S_1^4 \), \( S_2^2 \), and \( S_3^2 \)

- **Calculate \( S_1^4 \)**:
\[ S_1^4 = \left( \frac{n(n + 1)}{2} \right)^4 \]

- **Calculate \( S_2^2 \)**:
\[ S_2^2 = \left( \frac{n(n + 1)(2n + 1)}{6} \right)^2 \]

- **Calculate \( S_3^2 \)**:
\[ S_3^2 = \left( \left( \frac{n(n + 1)}{2} \right)^2 \right)^2 = \left( \frac{n(n + 1)}{2} \right)^4 \]

### Step 4: Substitute \( S_1^4 \), \( S_2^2 \), and \( S_3^2 \) back into the expression

Now substitute these back into the expression:

\[ \frac{\left( \frac{n(n + 1)}{2} \right)^4 \left( \frac{n(n + 1)(2n + 1)}{6} \right)^2 - \left( \frac{n(n + 1)(2n + 1)}{6} \right)^2 \left( \frac{n(n + 1)}{2} \right)^4}{\left( \frac{n(n + 1)}{2} \right)^2 + \left( \frac{n(n + 1)(2n + 1)}{6} \right)^2} \]

### Step 5: Simplify the expression

Notice that both terms in the numerator have a common factor \( S_2^2 \):

\[ S_2^2 \left( S_1^4 - S_3^2 \right) \]

Since \( S_1^4 \) and \( S_3^2 \) are equal:

\[ S_1^4 = S_3^2 \]

Thus, the numerator becomes:

\[ S_2^2 (S_1^4 - S_3^2) = S_2^2 \cdot 0 = 0 \]

### Step 6: Conclusion

The entire expression simplifies to:

\[ \frac{0}{S_1^2 + S_2^2} = 0 \]

Thus, the final answer is:

\[ \boxed{0} \]