Question

If p, q, r are in one geometric progression and a, b, c are in another geometric progression, then cp, bq and ar are in
1. A.P
2. H.P
3. G.P
4. None of these.

Answer 1

Hint: In this problem, we are given that p, q, r are in one geometric progression and a, b, c are in another geometric progression. We have to find the progression in which the terms cp, bq and ar are in. We can first find the common ratio for each of the terms, we can then multiply the terms and we should know that if we multiply the terms in geometric progression to some other terms in another geometric progression, the resulting terms are also in geometric progression.

Complete step by step solution:
Here we are given that p, q, r are in one geometric progression and a, b, c are in another geometric progression.
We can first find the common ratio for p, q, r.
We know that the common ratio is the ratio of the successive term and the preceding term.
Let \[{{R}_{1}}\] be the common ratio for p, q, r.
Common rations are,
\[\begin{align}
  & \Rightarrow {{R}_{1}}=\dfrac{q}{p} \\
 & \Rightarrow p=\dfrac{q}{{{R}_{1}}}......(1) \\
\end{align}\]
\[\begin{align}
  & \Rightarrow {{R}_{1}}=\dfrac{r}{q} \\
 & \Rightarrow q=\dfrac{r}{{{R}_{1}}}.......(2) \\
\end{align}\]
Let \[{{R}_{2}}\] be the common ratio for a, b, c.
Common ratios are,
\[\begin{align}
  & \Rightarrow {{R}_{2}}=\dfrac{b}{a} \\
 & \Rightarrow b=a{{R}_{2}}......(3) \\
\end{align}\]
\[\begin{align}
  & \Rightarrow {{R}_{2}}=\dfrac{c}{b} \\
 & \Rightarrow c=b{{R}_{2}}......(4) \\
\end{align}\]
We can now multiply (2) and (3), we get
\[\Rightarrow bq=ar\dfrac{{{R}_{2}}}{{{R}_{1}}}\] ……… (5)
We can now multiply (1) and (4), we get
\[\Rightarrow cp=bq\dfrac{{{R}_{2}}}{{{R}_{1}}}\] ………. (6)
We can see that from (5) and (6), cp, bq, ar are also in geometric progression if the terms p, q, r in one geometric progression are multiplied to the terms a, b, c in another geometric progression.
Therefore, the answer is option 3. G.P.

Note: We should always remember that if we multiply the terms in geometric progression to some other terms in another geometric progression, the resulting terms are also in geometric progression. We should also know that the common ratio is the ratio of the successive term and the preceding term.