First slide

Geometric progression

Question

If $a^{2} + b^{2}, ab + bc and b^{2} + c^{2}$ are in G.P., then a, b, c are in

Moderate

A

A.P
B

G.P
C

H.P
D

none of these

Solution

$\begin{array}{l} a^{2} + b^{2}, ab + bc, b^{2} + c^{2} are in G.P. \\ \Rightarrow (ab + bc)^{2} = (a^{2} + b^{2}) (b^{2} + c^{2}) \\ or a^{2} b^{2} + b^{2} c^{2} + 2 {ab}^{2} c = a^{2} b^{2} + a^{2} c^{2} + b^{2} c^{2} + b^{4} \\ or b^{4} + a^{2} c^{2} - 2 {ab}^{2} c = 0 \\ or {(b^{2} - ac)}^{2} = 0 \\ or b^{2} = ac \end{array}$
Hence, a, b, and c are in G.P.

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