Question

Find two consecutive numbers, the sum of whose squares are 365.

Answer 1

Given : Sum of the squares of two consecutive numbers is 365.

To Find : Those two consecutive numbers.

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Here, the sum of two consecutive numbers is given as 365.

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• Let the first consecutive number be x

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So, second consecutive number will be (x + 1)

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Now,

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• Square of first number = x²
• Square of second number = (x + 1)²

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$\bold{ \underline{According \: to \: the \:question \: : }} \:$

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$\sf:\implies \: (x) {}^{2} + (x + 1) { }^{2} = 365 \\ \\ \\ \sf:\implies \: x {}^{2} + x {}^{2} + 2x + 1 = 365 \\ \\ \\ \sf:\implies \: 2x {}^{2} + 2x + 1 = 365 \\ \\ \\ \sf:\implies \: 2x {}^{2} + 2x + 1 - 365 \\ \\ \\ \sf:\implies \: 2x {}^{2} + 2x - 364 = 0 \\ \\ \\ \sf:\implies \: x {}^{2} + x - 182 = 0 \\ \\ \\ \sf:\implies \: x {}^{2} + 14x - 13x - 182 = 0 \: \: \: \: \: \small(by \: splitting \:the \: middle \: term)\\ \\ \\ \sf:\implies \: x(x + 14) - 13(x + 14) = 0 \\ \\ \\ \sf:\implies \: (x +14)(x - 13) \\ \\ \\ \sf:\implies \: \underline{\boxed{\mathfrak\purple{x = 13}}} \: \bigstar \: \: \: \: \: \: \small(ignore \: negative \: value) \:$

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Now,

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• First consecutive number = 13
• Second consecutive number = 13 + 1 = 14

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$\sf\therefore{\underline{Hence, \: the \: two \: consecutive \: numbers \: are \: \bold{13} \: and \: \bold{14} \: respectively.}}$