If the surm 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 `16/5m` ,then m is equal to

Options

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

Explanation

To find the value of \( m \) in the given series, we need to first analyze the series and find the sum of the first ten terms. The series is:

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

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers to improper fractions:

- \( 1 \frac{3}{5} = \frac{8}{5} \)
- \( 2 \frac{2}{5} = \frac{12}{5} \)
- \( 3 \frac{1}{5} = \frac{16}{5} \)
- \( 4 = \frac{20}{5} \)
- \( 4 \frac{4}{5} = \frac{24}{5} \)

### Step 2: Identify the General Term
The general term of the series can be observed as:

\[ \left( \frac{4n + 4}{5} \right)^2 \]

Where \( n \) is the term number starting from 1.

### Step 3: Write the Sum of the First 10 Terms
The sum of the first 10 terms can be expressed as:

\[ S_{10} = \sum_{n=1}^{10} \left( \frac{4n + 4}{5} \right)^2 \]

This simplifies to:

\[ S_{10} = \frac{1}{25} \sum_{n=1}^{10} (4n + 4)^2 \]

### Step 4: Expand the Expression
Now we expand \( (4n + 4)^2 \):

\[ (4n + 4)^2 = 16n^2 + 32n + 16 \]

### Step 5: Sum the Series
Now we can write the sum:

\[ S_{10} = \frac{1}{25} \sum_{n=1}^{10} (16n^2 + 32n + 16) \]

This can be separated into three sums:

\[ S_{10} = \frac{1}{25} \left( 16 \sum_{n=1}^{10} n^2 + 32 \sum_{n=1}^{10} n + 16 \sum_{n=1}^{10} 1 \right) \]

### Step 6: Use Known Formulas for Sums
We use the formulas for the sums:

- \( \sum_{n=1}^{k} n^2 = \frac{k(k+1)(2k+1)}{6} \)
- \( \sum_{n=1}^{k} n = \frac{k(k+1)}{2} \)
- \( \sum_{n=1}^{k} 1 = k \)

For \( k = 10 \):

- \( \sum_{n=1}^{10} n^2 = \frac{10 \cdot 11 \cdot 21}{6} = 385 \)
- \( \sum_{n=1}^{10} n = \frac{10 \cdot 11}{2} = 55 \)
- \( \sum_{n=1}^{10} 1 = 10 \)

### Step 7: Substitute Back into the Equation
Now substituting these values back into the equation for \( S_{10} \):

\[ S_{10} = \frac{1}{25} \left( 16 \cdot 385 + 32 \cdot 55 + 16 \cdot 10 \right) \]

Calculating each term:

- \( 16 \cdot 385 = 6160 \)
- \( 32 \cdot 55 = 1760 \)
- \( 16 \cdot 10 = 160 \)

Thus,

\[ S_{10} = \frac{1}{25} (6160 + 1760 + 160) = \frac{1}{25} \cdot 8090 = \frac{8090}{25} \]

### Step 8: Set Equal to Given Expression
We are given that:

\[ S_{10} = \frac{16}{5} m \]

Setting the two expressions equal:

\[ \frac{8090}{25} = \frac{16}{5} m \]

### Step 9: Solve for \( m \)
Cross-multiplying gives:

\[ 8090 \cdot 5 = 16 \cdot 25 m \]

\[ 40450 = 400 m \]

Thus,

\[ m = \frac{40450}{400} = 101.125 \]

### Final Step: Conclusion
Since \( m \) must be an integer, we round it to \( 101 \).

So, the value of \( m \) is:

\[ \boxed{101} \]