Let m be the smallest odd positive iteger for which `1 + 2+ …..+` m is a square of an integer and let n be the smallest even positive integer for which `1 + 2+ ……+` n is a square of an integer. What is the value of m + n ?

Answer

To solve the problem, we need to find two integers: \( m \), the smallest odd positive integer such that the sum of the first \( m \) natural numbers is a perfect square, and \( n \), the smallest even positive integer such that the sum of the first \( n \) natural numbers is also a perfect square. Finally, we will compute \( m + n \).

### Step 1: Finding \( m \)

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

\[ S_m = \frac{m(m + 1)}{2} \]

We want \( S_m \) to be a perfect square. Therefore, we need to find the smallest odd positive integer \( m \) such that:

\[ \frac{m(m + 1)}{2} = k^2 \quad \text{for some integer } k \]

#### Testing odd integers for \( m \):

1. **For \( m = 1 \)**:
\[ S_1 = \frac{1(1 + 1)}{2} = \frac{1 \cdot 2}{2} = 1 \quad (1^2) \]
This is a perfect square.

Since 1 is the smallest odd positive integer, we have \( m = 1 \).

### Step 2: Finding \( n \)

Next, we need to find the smallest even positive integer \( n \) such that:

\[ \frac{n(n + 1)}{2} = k^2 \quad \text{for some integer } k \]

#### Testing even integers for \( n \):

1. **For \( n = 2 \)**:
\[ S_2 = \frac{2(2 + 1)}{2} = \frac{2 \cdot 3}{2} = 3 \quad (not\ a\ perfect\ square) \]

2. **For \( n = 4 \)**:
\[ S_4 = \frac{4(4 + 1)}{2} = \frac{4 \cdot 5}{2} = 10 \quad (not\ a\ perfect\ square) \]

3. **For \( n = 6 \)**:
\[ S_6 = \frac{6(6 + 1)}{2} = \frac{6 \cdot 7}{2} = 21 \quad (not\ a\ perfect\ square) \]

4. **For \( n = 8 \)**:
\[ S_8 = \frac{8(8 + 1)}{2} = \frac{8 \cdot 9}{2} = 36 \quad (6^2) \]
This is a perfect square.

Thus, the smallest even positive integer \( n \) is \( 8 \).

### Step 3: Calculating \( m + n \)

Now, we can calculate:

\[ m + n = 1 + 8 = 9 \]

### Final Answer:

The value of \( m + n \) is \( \boxed{9} \).

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