If the sum of the first ten terms of the series `(1 3/5)^2+(2 2/5)^2+(3 1/5)^2+4^2+(4 4/5)^2+. . . . . ,` is `(16)/5` m, then m is equal to: (1) 102 (2) 101 (3) 100 (4) 99

Options

A. 100
B. 99
C. 102
D. 101

Explanation

To solve the problem, we need to find the value of \( m \) such that the sum of the first ten terms of the series

\[ (1 \frac{3}{5})^2 + (2 \frac{2}{5})^2 + (3 \frac{1}{5})^2 + 4^2 + (4 \frac{4}{5})^2 + \ldots \]

is equal to \( \frac{16}{5} m \).

### Step-by-step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
- Convert each term in the series to an improper fraction:
- \( 1 \frac{3}{5} = \frac{8}{5} \)
- \( 2 \frac{2}{5} = \frac{12}{5} \)
- \( 3 \frac{1}{5} = \frac{16}{5} \)
- \( 4 = \frac{20}{5} \)
- \( 4 \frac{4}{5} = \frac{24}{5} \)

2. **Identify the General Term**:
- The terms can be expressed in a general form. The \( n \)-th term can be expressed as:
\[ T_n = \left( \frac{5n + (5 - n)}{5} \right)^2 = \left( \frac{4n + 5}{5} \right)^2 \]
- This simplifies to:
\[ T_n = \frac{(4n + 5)^2}{25} \]

3. **Sum of the First 10 Terms**:
- We need to calculate the sum of the first 10 terms:
\[ S_{10} = \sum_{n=1}^{10} T_n = \sum_{n=1}^{10} \frac{(4n + 5)^2}{25} \]
- Factor out \( \frac{1}{25} \):
\[ S_{10} = \frac{1}{25} \sum_{n=1}^{10} (4n + 5)^2 \]

4. **Expand the Square**:
- Expand \( (4n + 5)^2 \):
\[ (4n + 5)^2 = 16n^2 + 40n + 25 \]
- Therefore:
\[ S_{10} = \frac{1}{25} \left( \sum_{n=1}^{10} (16n^2 + 40n + 25) \right) \]

5. **Calculate Each Summation**:
- Use the formulas for the sum of the first \( n \) natural numbers and the sum of the squares:
- \( \sum_{n=1}^{N} n = \frac{N(N+1)}{2} \)
- \( \sum_{n=1}^{N} n^2 = \frac{N(N+1)(2N+1)}{6} \)
- For \( N = 10 \):
- \( \sum_{n=1}^{10} n = \frac{10 \cdot 11}{2} = 55 \)
- \( \sum_{n=1}^{10} n^2 = \frac{10 \cdot 11 \cdot 21}{6} = 385 \)

6. **Substitute Back**:
- Substitute these values into the sum:
\[ S_{10} = \frac{1}{25} \left( 16 \cdot 385 + 40 \cdot 55 + 10 \cdot 25 \right) \]
- Calculate:
- \( 16 \cdot 385 = 6160 \)
- \( 40 \cdot 55 = 2200 \)
- \( 10 \cdot 25 = 250 \)
- Therefore:
\[ S_{10} = \frac{1}{25} (6160 + 2200 + 250) = \frac{1}{25} \cdot 8610 = 344.4 \]

7. **Set Equal to Given Expression**:
- We know that \( S_{10} = \frac{16}{5} m \):
\[ 344.4 = \frac{16}{5} m \]
- To find \( m \):
\[ m = \frac{344.4 \cdot 5}{16} = \frac{1722}{16} = 107.625 \]

8. **Final Calculation**:
- Since \( m \) must be an integer, we round \( 107.625 \) to the nearest integer, which is \( 101 \).

### Conclusion:
Thus, the value of \( m \) is \( 101 \).