Progression (Sequence and Series)HardBloom L3
Question
If $a,b,c$ be in A.P. and $a^{2},{\text{ }b}^{2},c^{2}$ be in H.P., then which of the following(s) may be true?
Options
A.$\frac{a}{2},\text{ }b,c$ are in G.P.
B.$a, - \frac{b}{2},c$ are in G.P.
C.$- \frac{a}{2},\text{ }b,c$ are in G.P.
D.$a = b = c$
Solution
Sol. [Key Concept: Numbers in A.P. and in H.P. ⇒ the related quantities are in G.P.]
$$b^{2} = \frac{2a^{2}c^{2}}{a^{2} + c^{2}} \Rightarrow \left( (a + c)^{2} - 2ac \right)b^{2} = 2a^{2}c^{2}$$
$$\begin{matrix} \Rightarrow & 2{\text{ }b}^{4} - {ac}^{2} - a^{2}c^{2} = 0 = \left( 2{\text{ }b}^{2} + ac \right)\left( b^{2} - ac \right) \\ \Rightarrow & b^{2} = \left( - \frac{a}{2} \right)c\ \text{~}\text{or}\text{~}\ b^{2} = ac \end{matrix}$$
✘ (a) not forced
✘ (b) not forced
✔ (c) $b^{2 =}ac$ with the AP condition
✔ (d) forces $a = b = c$
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