Question

If a, b, c is in A.P., prove that:

a² + c² + 4ac = 2 (ab + bc + ca)

Answer 1

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

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

Consider RHS:
2 (ab + bc + ca)

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

\[ \Rightarrow 2\left\{ a\left( \frac{a + c}{2} \right) + c\left( \frac{a + c}{2} \right) + ac \right\}\]

\[ \Rightarrow 2\left\{ \frac{a^2 + ac + ac + c^2 + 2ac}{2} \right\}\]

\[ \Rightarrow a^2 + 4ac + c^2 \]

\[\text { Hence, proved } .\]