Question

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

Answer 1

Solution -

Let numbers be $a,a+1,a+2,----a+7$

Their square sum $=a^2+(a+1{)}^2+(a+2{)}^2+----(a+7{)}^2$

$=8a^2+140+56a$

Now LHS

$a^2+d^2+f^2+g^2=a^2+(a+3{)}^2+(a+5{)}^2+(a+6{)}^2$

$=4a^2+70+28a$

RHS, $b^2+c^2+e^2+h^2=(a+1{)}^2+(a+2{)}^2+(a+4{)}^2+(a+7{)}^2$

$=4a^2+70+28a$

LHS = RHS