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Question
If a, b, c is in A.P., prove that:
(a − c)2 = 4 (a − b) (b − c)
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Solution
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.
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