Question

If a, b, c is in A.P., prove that:

 (a − c)² = 4 (a − b) (b − c)

Answer 1

Since a, b, c are in A.P., we have:
    2b = a+c

\[\Rightarrow\] b = \[\frac{a + c}{2}\]

Consider RHS:
4 (a − b) (b − c)

\[\text { Substituting b } = \frac{a + c}{2}: \]

\[ \Rightarrow 4\left\{ a - \frac{a + c}{2} \right\}\left\{ \frac{a + c}{2} - c \right\}\]

\[ \Rightarrow 4\left\{ \frac{2a - a - c}{2} \right\}\left\{ \frac{a + c - 2c}{2} \right\}\]

\[ \Rightarrow \left( a - c \right)\left( a - c \right)\]

\[ \Rightarrow \left( a - c \right)^2\]

Hence, proved.