1595 is the sum of the square of three consecutive odd numbers. Find the numbers.

Options

A. 19,21,23
B. 17,19,21
C. 21,23,25
D. 23,25,27

Answer

To solve the problem of finding three consecutive odd numbers whose squares sum up to 1595, we can follow these steps:

### Step 1: Define the Odd Numbers
Let the first odd number be \( a \). Therefore, the three consecutive odd numbers can be expressed as:
- First number: \( a \)
- Second number: \( a + 2 \)
- Third number: \( a + 4 \)

### Step 2: Set Up the Equation
According to the problem, the sum of the squares of these three numbers equals 1595. We can write this as:
\[ a^2 + (a + 2)^2 + (a + 4)^2 = 1595 \]

### Step 3: Expand the Squares
Now, we will expand the squares:
\[ a^2 + (a^2 + 4a + 4) + (a^2 + 8a + 16) = 1595 \]

### Step 4: Combine Like Terms
Combining the terms gives us:
\[ a^2 + a^2 + 4a + 4 + a^2 + 8a + 16 = 1595 \]
\[ 3a^2 + 12a + 20 = 1595 \]

### Step 5: Rearrange the Equation
Now, we will move 1595 to the left side of the equation:
\[ 3a^2 + 12a + 20 - 1595 = 0 \]
\[ 3a^2 + 12a - 1575 = 0 \]

### Step 6: Simplify the Equation
To simplify the equation, we can divide all terms by 3:
\[ a^2 + 4a - 525 = 0 \]

### Step 7: Factor the Quadratic Equation
Next, we will factor the quadratic equation. We need two numbers that multiply to -525 and add to 4. The numbers are 25 and -21:
\[ (a + 25)(a - 21) = 0 \]

### Step 8: Solve for \( a \)
Setting each factor to zero gives us:
1. \( a + 25 = 0 \) → \( a = -25 \) (not valid since we need odd positive numbers)
2. \( a - 21 = 0 \) → \( a = 21 \)

### Step 9: Find the Three Odd Numbers
Now that we have \( a = 21 \), we can find the three consecutive odd numbers:
- First number: \( 21 \)
- Second number: \( 21 + 2 = 23 \)
- Third number: \( 21 + 4 = 25 \)

### Conclusion
The three consecutive odd numbers are **21, 23, and 25**.

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