Question

The sum of the squares of five consecutive natural numbers is 1455. Find the numbers.

Hint:

When dealing with consecutive numbers, always take the middle term as \(x\) to simplify square or cube-based equations.

Answer 1

Step 1: Assume the numbers.
Let the five consecutive natural numbers be: \[ x - 2, \; x - 1, \; x, \; x + 1, \; x + 2 \] Step 2: Write their squares and take the sum.
\[ (x - 2)^2 + (x - 1)^2 + x^2 + (x + 1)^2 + (x + 2)^2 = 1455 \] Step 3: Expand and simplify.
\[ (x^2 - 4x + 4) + (x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1) + (x^2 + 4x + 4) = 1455 \] \[ 5x^2 + ( -4x - 2x + 2x + 4x ) + (4 + 1 + 1 + 4) = 1455 \] \[ 5x^2 + 0x + 10 = 1455 \] Step 4: Solve for \(x\).
\[ 5x^2 = 1455 - 10 = 1445 \] \[ x^2 = 289 \] \[ x = 17 \] Step 5: Write the five consecutive numbers.
\[ x - 2 = 15, \; x - 1 = 16, \; x = 17, \; x + 1 = 18, \; x + 2 = 19 \] Step 6: Conclusion.
Hence, the required five consecutive natural numbers are \(15, 16, 17, 18, 19.\)
Final Answer: \[ \boxed{15, 16, 17, 18, 19} \]