Find the sum of the series `31^3+32^3++50^3dot`

Explanation

To find the sum of the series \(31^3 + 32^3 + 33^3 + \ldots + 50^3\), we can use the formula for the sum of cubes of the first \(n\) natural numbers, which is given by:

\[ S_n = \left(\frac{n(n+1)}{2}\right)^2 \]

### Step-by-step Solution:

1. **Identify the range of the series**:
We need to find the sum of cubes from \(31\) to \(50\).

2. **Express the sum in terms of known sums**:
We can express the sum \(31^3 + 32^3 + \ldots + 50^3\) as:
\[ S_{50} - S_{30} \]
where \(S_n\) is the sum of cubes of the first \(n\) natural numbers.

3. **Calculate \(S_{50}\)**:
Using the formula for the sum of cubes:
\[ S_{50} = \left(\frac{50 \times 51}{2}\right)^2 \]
First, calculate \(\frac{50 \times 51}{2}\):
\[ \frac{50 \times 51}{2} = \frac{2550}{2} = 1275 \]
Now, square this result:
\[ S_{50} = 1275^2 = 1625625 \]

4. **Calculate \(S_{30}\)**:
Similarly, we calculate:
\[ S_{30} = \left(\frac{30 \times 31}{2}\right)^2 \]
First, calculate \(\frac{30 \times 31}{2}\):
\[ \frac{30 \times 31}{2} = \frac{930}{2} = 465 \]
Now, square this result:
\[ S_{30} = 465^2 = 216225 \]

5. **Find the sum of the series**:
Now, substitute \(S_{50}\) and \(S_{30}\) into the expression:
\[ S_{31 \text{ to } 50} = S_{50} - S_{30} = 1625625 - 216225 \]
Calculate the difference:
\[ S_{31 \text{ to } 50} = 1409400 \]

### Final Answer:
The sum of the series \(31^3 + 32^3 + \ldots + 50^3\) is \(1409400\).