Question

If a, b, c are in AP prove that $(a-c{)}^2=4(b^2-ac)$.

Answer 1

$a,b,c$ are in $AP$

$b-a=c-b$

$2b=a+c$

$LHS$

$(a-c{)}^2=a^2-2ac+c^2$

$RHS$

$4(b^2-ac)$

$=(2b{)}^2-4ac$

$=(a+c{)}^2-4ac$

$=(a-c{)}^2$

$LHS=RHS$