Let K is a positive Integer such that `36 +K , 300 + K , 596 + K` are the squares of teree consecutive terms of an arithmetic progression. Find K.

Answer

To solve the problem, we need to find the positive integer \( K \) such that \( 36 + K \), \( 300 + K \), and \( 596 + K \) are the squares of three consecutive terms of an arithmetic progression (AP).

Let's denote the three consecutive terms of the AP as \( A - D \), \( A \), and \( A + D \).

### Step 1: Set up the equations
From the problem, we can write the following equations based on the squares of the terms:
1. \( (A - D)^2 = 36 + K \)
2. \( A^2 = 300 + K \)
3. \( (A + D)^2 = 596 + K \)

### Step 2: Expand the equations
Expanding the first and third equations using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \) and \( (a + b)^2 = a^2 + 2ab + b^2 \):
1. \( A^2 - 2AD + D^2 = 36 + K \)
2. \( A^2 = 300 + K \)
3. \( A^2 + 2AD + D^2 = 596 + K \)

### Step 3: Rearrange the equations
Now, we can rearrange the first and third equations:
1. \( A^2 + D^2 - 2AD = 36 + K \) (Equation 1)
2. \( A^2 + D^2 + 2AD = 596 + K \) (Equation 2)

### Step 4: Subtract the equations
Next, we subtract Equation 1 from Equation 2:
\[ (A^2 + D^2 + 2AD) - (A^2 + D^2 - 2AD) = (596 + K) - (36 + K) \]
This simplifies to:
\[ 4AD = 560 \]
Thus, we find:
\[ AD = 140 \]

### Step 5: Substitute \( AD \) back into Equation 1
Now, we substitute \( AD = 140 \) back into Equation 1:
\[ A^2 + D^2 - 2(140) = 36 + K \]
This simplifies to:
\[ A^2 + D^2 - 280 = 36 + K \]
So we have:
\[ A^2 + D^2 = K + 316 \quad \text{(Equation 3)} \]

### Step 6: Use Equation 2 to find \( D^2 \)
From Equation 2, we know:
\[ A^2 = 300 + K \]
Substituting \( A^2 \) into Equation 3:
\[ (300 + K) + D^2 = K + 316 \]
This leads to:
\[ D^2 = 316 - 300 = 16 \]
Thus, we find:
\[ D = \pm 4 \]

### Step 7: Find \( A \)
Using \( AD = 140 \):
1. If \( D = 4 \):
\[ A \cdot 4 = 140 \implies A = 35 \]
2. If \( D = -4 \):
\[ A \cdot (-4) = 140 \implies A = -35 \]
Since \( A \) must be positive, we take \( A = 35 \).

### Step 8: Find \( K \)
Now substituting \( A = 35 \) back into Equation 2:
\[ A^2 = 300 + K \implies 35^2 = 300 + K \]
Calculating \( 35^2 = 1225 \):
\[ 1225 = 300 + K \implies K = 1225 - 300 = 925 \]

Thus, the value of \( K \) is \( \boxed{925} \).