Let `S_(k) = (1+2+3+...+k)/(k)`. If `S_(1)^(2) + s_(2)^(2) +...+S_(10)^(2) = (5)/(12)A`, then A is equal to

Options

A. 303
B. 283
C. 156
D. 301

Answer

To solve the problem, we need to find the value of \( A \) given that

\[ S_1^2 + S_2^2 + \ldots + S_{10}^2 = \frac{5}{12} A \]

where

\[ S_k = \frac{1 + 2 + 3 + \ldots + k}{k} \]

### Step 1: Calculate \( S_k \)

The sum of the first \( k \) natural numbers is given by the formula:

\[ 1 + 2 + 3 + \ldots + k = \frac{k(k + 1)}{2} \]

Thus, we can express \( S_k \) as:

\[ S_k = \frac{\frac{k(k + 1)}{2}}{k} = \frac{k + 1}{2} \]

### Step 2: Calculate \( S_k^2 \)

Now we find \( S_k^2 \):

\[ S_k^2 = \left(\frac{k + 1}{2}\right)^2 = \frac{(k + 1)^2}{4} \]

### Step 3: Calculate \( S_1^2 + S_2^2 + \ldots + S_{10}^2 \)

Now we need to compute:

\[ S_1^2 + S_2^2 + \ldots + S_{10}^2 = \sum_{k=1}^{10} S_k^2 = \sum_{k=1}^{10} \frac{(k + 1)^2}{4} \]

Factoring out \( \frac{1}{4} \):

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

### Step 4: Simplify the summation

The expression \( (k + 1)^2 \) can be rewritten as:

\[ (k + 1)^2 = k^2 + 2k + 1 \]

Thus,

\[ \sum_{k=1}^{10} (k + 1)^2 = \sum_{k=1}^{10} k^2 + \sum_{k=1}^{10} 2k + \sum_{k=1}^{10} 1 \]

Using the formulas for these sums:

- The sum of the first \( n \) natural numbers is:

\[ \sum_{k=1}^{n} k = \frac{n(n + 1)}{2} \]

- The sum of the squares of the first \( n \) natural numbers is:

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

For \( n = 10 \):

1. \( \sum_{k=1}^{10} k^2 = \frac{10(11)(21)}{6} = 385 \)
2. \( \sum_{k=1}^{10} k = \frac{10(11)}{2} = 55 \)
3. \( \sum_{k=1}^{10} 1 = 10 \)

Combining these:

\[ \sum_{k=1}^{10} (k + 1)^2 = 385 + 2 \cdot 55 + 10 = 385 + 110 + 10 = 505 \]

### Step 5: Substitute back into the equation

Now substituting back into our equation for \( S_1^2 + S_2^2 + \ldots + S_{10}^2 \):

\[ S_1^2 + S_2^2 + \ldots + S_{10}^2 = \frac{1}{4} \cdot 505 = \frac{505}{4} \]

### Step 6: Set up the equation for \( A \)

Now we have:

\[ \frac{505}{4} = \frac{5}{12} A \]

### Step 7: Solve for \( A \)

Cross-multiplying gives:

\[ 505 \cdot 12 = 5 \cdot 4 A \]

\[ 6060 = 20A \]

\[ A = \frac{6060}{20} = 303 \]

### Final Answer

Thus, the value of \( A \) is:

\[ \boxed{303} \]