Mathematics
Out of three consecutive positive integers, the middle number is p. If three times the square of the largest is greater than the sum of the squares of the other two numbers by 67; calculate the value of p.
Answer
Let the numbers be (p - 1), p and (p + 1).
According to question,
⇒ 3(p + 1)2 = (p - 1)2 + p2 + 67
⇒ 3(p2 + 1 + 2p) = (p2 + 1 - 2p) + p2 + 67
⇒ 3p2 + 3 + 6p = 2p2 - 2p + 68
⇒ 3p2 - 2p2 + 6p + 2p + 3 - 68 = 0
⇒ p2 + 8p - 65 = 0
⇒ p2 + 13p - 5p - 65 = 0
⇒ p(p + 13) - 5(p + 13) = 0
⇒ (p - 5)(p + 13) = 0
⇒ p - 5 = 0 or p + 13 = 0
⇒ p = 5 or p = -13
Since, numbers are natural numbers,
∴ p ≠ -13.
∴ p - 1 = 4 and p + 1 = 6.
Hence, p = 5.
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