First slide

Geometric progression

Question

Let $a_{1}, a_{2}, a_{3}, a_{4} and a_{5}$ be such that $a_{1}, a_{2} and a_{3}$ are in A.P , $a_{2}, a_{3} and a_{4}$ are in G.P.; and $a_{3}, a_{4} and a_{5}$ are in H.P. Then $a_{1}, a_{3} and a_{5}$ are in

Moderate

A

G.P
B

A.P
C

H.P
D

None of these

Solution

We have,
$\begin{aligned} a_{1}, a_{2}, a_{3} are in A.P. \Rightarrow 2 a_{2} = a_{1} + a_{3} - - - - (i) \\ a_{2}, a_{3}, a_{4} are in G.P. \Rightarrow {a_{3}}^{2} = a_{2} a_{4} - - - - - (ii) \\ a_{3}, a_{4}, a_{5} are in H.P. \Rightarrow a_{4} = \frac{2 a_{3} a_{5}}{a_{3} + a_{5}} - - - - - (iii) \end{aligned}$
Putting $a_{2} = \frac{a_{1} + a_{3}}{2}$ and $a_{4} = \frac{2 a_{3} a_{5}}{a_{3} + a_{5}}$ in (ii), we get
$\begin{aligned} a_{3}^{2} & = \frac{a_{1} + a_{3}}{2} \times \frac{2 a_{3} a_{5}}{a_{3} + a_{5}} \\ \Rightarrow a_{3}^{2} & = a_{1} a_{5} \Rightarrow a_{1}, a_{3}, a_{5} are in G.P. \end{aligned}$

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