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Natural Numbers

If a, b, c,...

Question

If $a, b, c, d, e, f, g, h$ are eight consecutive natural numbers, then prove that
$a^{2} + d^{2} + f^{2} + g^{2} = b^{2} + c^{2} + e^{2} + h^{2}$

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Solution

Solution -
Let numbers be $a, a + 1, a + 2, - - - - a + 7$
Their square sum $= a^{2} + (a + 1)^{2} + (a + 2)^{2} + - - - - (a + 7)^{2}$
$= 8 a^{2} + 140 + 56 a$
Now LHS
$a^{2} + d^{2} + f^{2} + g^{2} = a^{2} + (a + 3)^{2} + (a + 5)^{2} + (a + 6)^{2}$
$= 4 a^{2} + 70 + 28 a$
RHS, $b^{2} + c^{2} + e^{2} + h^{2} = (a + 1)^{2} + (a + 2)^{2} + (a + 4)^{2} + (a + 7)^{2}$
$= 4 a^{2} + 70 + 28 a$
LHS = RHS

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