Let `n inN` and k be an integer `ge0` such that
`S_(k)(n)=1^(k)+2^(k)+3^(k)+ . . . +n^(k)`
Statement-1: `S_(4)(n)=(n)/(30)(n+1)(2n+1)(3n^(2)+3n+1)`
Statement -2: `.^(k+1)C_(1)S_(k)(n)+.^(k+1)C_(2)S_(k-1)(n)+ . . . +.^(k+1)C_(k)S_(1)(n)+.^(k+1)C_(k+1)S_(0)(n)=(n+1)^(k+1)-1`

Options

A. Statement -1 is true, Statement -2 is True, Statement -2 is a correct explanation for Statement for Statement -1.
B. Statement -1 is true, Statement -2 is True, Statement -2 is not a correct explanation for Statement for Statement -1.
C. Statement -1 is true, Statement -2 is False.
D. Statement -1 is False, Statement -2 is True.

Explanation

To solve the question, we need to analyze both statements provided and determine their validity.

### Step 1: Analyze Statement 2
Statement 2 states that:
\[ \binom{k+1}{1} S_k(n) + \binom{k+1}{2} S_{k-1}(n) + \ldots + \binom{k+1}{k} S_1(n) + \binom{k+1}{k+1} S_0(n) = (n+1)^{k+1} - 1 \]
This is a known identity in combinatorial mathematics, which can be derived from the binomial theorem.

To verify this, we can use the binomial expansion of \((r + 1)^{k+1}\) and \((r)^{k+1}\):
- The left-hand side can be seen as the expansion of \((r + 1)^{k+1} - r^{k+1}\) using the binomial theorem, which gives us the desired result.

Thus, Statement 2 is **true**.

### Step 2: Analyze Statement 1
Statement 1 states that:
\[ S_4(n) = \frac{n}{30} (n + 1)(2n + 1)(3n^2 + 3n + 1) \]
To verify this, we can use the result from Statement 2. We will replace \(k\) with \(4\) in Statement 2:
\[ \binom{5}{1} S_4(n) + \binom{5}{2} S_3(n) + \binom{5}{3} S_2(n) + \binom{5}{4} S_1(n) + \binom{5}{5} S_0(n) = (n + 1)^5 - 1 \]
Calculating the right-hand side:
\[ (n + 1)^5 - 1 = n^5 + 5n^4 + 10n^3 + 10n^2 + 5n \]

Now we need to express \(S_4(n)\), \(S_3(n)\), \(S_2(n)\), \(S_1(n)\), and \(S_0(n)\) using their respective formulas:
- \(S_0(n) = n\)
- \(S_1(n) = \frac{n(n + 1)}{2}\)
- \(S_2(n) = \frac{n(n + 1)(2n + 1)}{6}\)
- \(S_3(n) = \left(\frac{n(n + 1)}{2}\right)^2\)

Substituting these into the left-hand side and simplifying will lead us to the expression for \(S_4(n)\).

After performing the calculations, we find that the expression does not match the form given in Statement 1. Thus, Statement 1 is **not true**.

### Conclusion
- **Statement 1** is **false**.
- **Statement 2** is **true**.