A) A. P.
B) H. P.
C) G. P.
D) None of these
Correct Answer: A
Solution :
\[\cos A+2\cos B+\cos C=2\] Þ \[\cos A+\cos C=2(1-\cos B)\] Þ \[2\cos \frac{A+C}{2}\cos \frac{A-C}{2}=4{{\sin }^{2}}\frac{B}{2}\] Þ \[2\cos \left( \frac{A-C}{2} \right)=4\sin \frac{B}{2}\] Þ \[2\cos \frac{B}{2}\cos \left( \frac{A-C}{2} \right)=2\left( 2\sin \frac{B}{2}\cos \frac{B}{2} \right)\] Þ \[2\sin \,\left( \frac{A+C}{2} \right)\cos \left( \frac{A-C}{2} \right)=2\left( 2\sin \frac{B}{2}\cos \frac{B}{2} \right)\] Þ \[\sin A+\sin C=2\sin B\Rightarrow a+c=2b\] Þ a, b, c are in A.P.
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