If `S=1(25)+2(24)+3(23)+…………..+24(2)+25(1)` then the value of `(S)/(900)` is equal to

Answer

To solve the problem, we need to evaluate the sum \( S = 1(25) + 2(24) + 3(23) + \ldots + 24(2) + 25(1) \) and then find the value of \( \frac{S}{900} \).

### Step-by-Step Solution:

1. **Identify the Pattern**:
The series can be represented as:
\[ S = \sum_{n=1}^{25} n(26 - n) \]
Here, the first term \( n \) is increasing from 1 to 25, and the second term \( 26 - n \) is decreasing from 25 to 1.

2. **Expand the Summation**:
We can expand the summation:
\[ S = \sum_{n=1}^{25} (26n - n^2) \]
This can be separated into two sums:
\[ S = 26 \sum_{n=1}^{25} n - \sum_{n=1}^{25} n^2 \]

3. **Calculate the First Sum**:
The sum of the first \( n \) natural numbers is given by the formula:
\[ \sum_{n=1}^{N} n = \frac{N(N + 1)}{2} \]
For \( N = 25 \):
\[ \sum_{n=1}^{25} n = \frac{25 \cdot 26}{2} = 325 \]

4. **Calculate the Second Sum**:
The sum of the squares of the first \( n \) natural numbers is given by the formula:
\[ \sum_{n=1}^{N} n^2 = \frac{N(N + 1)(2N + 1)}{6} \]
For \( N = 25 \):
\[ \sum_{n=1}^{25} n^2 = \frac{25 \cdot 26 \cdot 51}{6} = 5525 \]

5. **Substitute Back into the Expression for \( S \)**:
Now substituting these values back into the expression for \( S \):
\[ S = 26 \cdot 325 - 5525 \]
Calculate \( 26 \cdot 325 \):
\[ 26 \cdot 325 = 8450 \]
Thus,
\[ S = 8450 - 5525 = 2925 \]

6. **Calculate \( \frac{S}{900} \)**:
Finally, we need to find:
\[ \frac{S}{900} = \frac{2925}{900} \]
Simplifying this gives:
\[ \frac{2925}{900} = 3.25 \]

### Final Answer:
The value of \( \frac{S}{900} \) is \( 3.25 \).