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The Sum of the Squares of Three Consecutive Natural Numbers is 2030. What is the Middle Number?

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Explanation

  • The sum of the squares of three consecutive natural numbers is 2030.
  • The middle number can be figure out in this way.

Let suppose the three consecutive natural numbers are y, y + 1 and y + 2.

The sum of y2, (y + 1)2 and (y + 2)2 is 2030.

y2 + (y + 1)2 + (y + 2)2 = 2030 ________ (i)

By simplifying equation (i), we can easily figure out the value of y (y = 25) and the middle number (y + 1 = 25 + 1 = 26).

To Find

Middle Number = ?

Solution

Let suppose

Numbers are y, y + 1 and y + 2.

According to the given conditions

y2 + (y + 1)2 + (y + 2)2 = 2030

y2 + y2 + 1 + 2y + y2 + 4 + 4y = 2030

3y2 + 6y + 5 = 2030

3y2 + 6y + 5 – 2030 = 0

3y2 + 6y – 2025 = 0

y2 + 2y – 675 = 0

y2 + 27y – 25y – 675 = 0

y(y + 27) – 25(y + 27) = 0

(y + 27)(y – 25) = 0

y + 27 = 0           &                y – 25 = 0

y = -27 (not possible)           &                y = 25 (possible)

Middle Number = 25 + 1 = 26 answer

Conclusion

The sum of the squares of three consecutive natural numbers is 2030. The middle number is 26.

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