Statement 1 :
The sum of the series 1+(1+2+4)+(4+6+9)+(9+12+16)+….+(361 +380 +400) is 8000
Statement 1:
`Sigma_(k=1)^(n) (k^3-(k-1)^3)=n^3`, for any natural number n.

Options

A. Statement 1 is fasle ,statement 2 is true
B. Statement 1 is true ,statement 2 is true , statement 2 is a correct explanation for statement 1.
C. Statement 1 is true, statements 2 is true statement 2 is not a correct explanation for statement 1
D. Statement 1 is true, statement 2 is false

Explanation

To solve the problem, we need to analyze the two statements provided:

**Statement 1**: The sum of the series \(1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \ldots + (361 + 380 + 400)\) is 8000.

**Statement 2**: \(\Sigma_{k=1}^{n} (k^3 - (k-1)^3) = n^3\) for any natural number \(n\).

### Step-by-Step Solution

**Step 1: Understand the Series in Statement 1**

The series given in Statement 1 can be broken down into groups:
- The first group is \(1\).
- The second group is \(1 + 2 + 4\).
- The third group is \(4 + 6 + 9\).
- The fourth group is \(9 + 12 + 16\).
- The last group is \(361 + 380 + 400\).

We need to identify a pattern in these groups.

**Hint for Step 1**: Look for a pattern in how the numbers are grouped and how they progress.

**Step 2: Identify the Pattern**

The first term is \(1\), the second group sums to \(7\) (which is \(1 + 2 + 4\)), the third group sums to \(19\) (which is \(4 + 6 + 9\)), and so on.

Notice that each group seems to consist of consecutive perfect squares:
- The first group has \(1^2\).
- The second group has \(1^2 + 2^2\).
- The third group has \(2^2 + 3^2\).
- The fourth group has \(3^2 + 4^2\).
- The last group has \(19^2 + 20^2\).

**Hint for Step 2**: Try to express each group in terms of squares or cubes.

**Step 3: Calculate the Sum of Each Group**

We can calculate the sum of each group:
- First group: \(1 = 1\)
- Second group: \(1 + 2 + 4 = 7\)
- Third group: \(4 + 6 + 9 = 19\)
- Fourth group: \(9 + 12 + 16 = 37\)
- Last group: \(361 + 380 + 400 = 1141\)

Now, we need to sum these results.

**Hint for Step 3**: Add the results of each group carefully.

**Step 4: Calculate the Total Sum**

Now, let's sum the values:
\[ 1 + 7 + 19 + 37 + 1141 = 1205 \]
This does not equal 8000, indicating that Statement 1 is false.

**Hint for Step 4**: Ensure you add all the groups correctly and check your calculations.

**Step 5: Analyze Statement 2**

For Statement 2, we can use the identity:
\[ k^3 - (k-1)^3 = 3k^2 - 3k + 1 \]
This represents the difference of cubes. We can sum this from \(k=1\) to \(n\):
\[ \Sigma_{k=1}^{n} (k^3 - (k-1)^3) = n^3 \]
This is a well-known result in mathematics.

**Hint for Step 5**: Review the properties of summation and cubes to understand why this identity holds.

### Conclusion

- **Statement 1** is false as the calculated sum does not equal 8000.
- **Statement 2** is true as it correctly states the summation of the difference of cubes.

### Final Answer

- Statement 1: False
- Statement 2: True