Question

The sum of the squares of three consecutive odd numbers is 2531. The numbers are :

• A. 27, 29 and 31
• B. 28, 30 and 32
• C. 29, 30 and 31
• D. 29, 31 and 32

Hint:

For problems involving consecutive odd or even numbers, representing them algebraically as \( x, x+2, x+4, \ldots \) can help in setting up equations. Often, testing the given options can be a quicker way to find the solution, especially if the numbers are relatively small.

Answer 1

The Correct Option is A

Step 1: Understand the properties of consecutive odd numbers.
Consecutive odd numbers differ by 2. If the first odd number is \( x \), the next two are \( x + 2 \) and \( x + 4 \). Step 2: Set up the equation based on the given information.
The sum of the squares of these three consecutive odd numbers is 2531. $$x^2 + (x + 2)^2 + (x + 4)^2 = 2531$$ Step 3: Expand the squares.
$$x^2 + (x^2 + 4x + 4) + (x^2 + 8x + 16) = 2531$$ Step 4: Combine like terms.
$$3x^2 + 12x + 20 = 2531$$ Step 5: Rearrange the equation into a quadratic form.
$$3x^2 + 12x + 20 - 2531 = 0$$ $$3x^2 + 12x - 2511 = 0$$ Step 6: Solve the quadratic equation. This can be done using the quadratic formula or by testing the options.
Lets test the options:
Option (1): 27, 29 and 31
$$27^2 + 29^2 + 31^2 = 729 + 841 + 961 = 2531$$ This matches the given sum. Option (2): 28, 30 and 32
These are consecutive even numbers, so this option is incorrect. Option (3): 29, 30 and 31
These are not all odd numbers, so this option is incorrect. Option (4): 29, 31 and 32
These are not all odd numbers, so this option is incorrect. Since option (1) satisfies the condition, it is the correct set of numbers.