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The sum of the squares of three consecutive natural numbers is 2030. Then, what is the middle number?
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- 25
- 26
- 27
- 28
Correct Option: B
Let us assume the three consecutive natural numbers be P, P + 1 and P + 2.
∴ According to the given question,
P2 + (P + 1)2 + (P + 2)2 = 2030
P2 + P2 + 2P + 1 + P2 + 4P + 4 = 2030
⇒ 3P2 + 6P + 5 = 2030
⇒ 3P2 + 6P − 2025 = 0
⇒ P2 + 2P – 675 = 0
⇒ P2 + 27P – 25P – 675 = 0
⇒ P (P + 27) – 25 (P+ 27) = 0
⇒ (P – 25) (P + 27) = 0
∴ P = 25 and – 27
∴ Required number = P + 1 = 25 + 1 = 26