Correct option is (D) a = b = c
Given that a, b, c are in AP and also in G.P.
\(\therefore a+c=2b\) ______________(1)
and \(ac=b^2\) ______________(2)
\(\Rightarrow ac=(\frac{a+c}2)^2\) (From (1))
\(\Rightarrow ac=\frac{a^2+c^2+2ac}4\)
\(\Rightarrow a^2+c^2+2ac=4ac\)
\(\Rightarrow a^2+c^2-2ac=0\)
\(\Rightarrow(a-c)^2=0\)
\(\Rightarrow a-c=0\)
\(\Rightarrow a=c\)
Put a = c in equation (1), we get
a+a = 2b
\(\Rightarrow2b=2a\)
\(\Rightarrow b=a\)
Hence, a = b = c
