If p, q, r are in one geometric progression and a, b, c are in another geometric progression, then ap, bq, cr are in

Concept:

In a geometric progression, the ratio between 2 consecutive terms remains constant.

Calculation:

Considering G.P of p, q, r let the common ratio be k₁.

\(\frac{{\rm{r}}}{{\rm{q}}} = {{\rm{k}}_1}{\rm{\;}};{\rm{\;}}\frac{{\rm{q}}}{{\rm{p}}} = {{\rm{k}}_1}{\rm{\;}}\)

\( \Rightarrow {\rm{q}} = {{\rm{k}}_1}{\rm{p\;}};{\rm{\;r}} = {{\rm{k}}_1}{\rm{q}} = {\rm{k}}_1^2{\rm{p\;}}\)

Considering G.P of a, b, c, let the common ratio be k₁.

\(\frac{{\rm{c}}}{{\rm{b}}} = {{\rm{k}}_2}{\rm{\;}};{\rm{\;}}\frac{{\rm{b}}}{{\rm{a}}} = {{\rm{k}}_2}{\rm{\;}}\)

\( \Rightarrow {\rm{b}} = {{\rm{k}}_2}{\rm{a\;}};{\rm{\;c}} = {{\rm{k}}_2}{\rm{b}} = {\rm{k}}_2^2{\rm{a\;}}\)

On multiplying both:

\({\rm{bq}} = {{\rm{k}}_2}{{\rm{k}}_1}{\rm{ap\;}};{\rm{\;cr}} = {{\rm{k}}_1}{{\rm{k}}_2}{\rm{bq}} = {\left( {{{\rm{k}}_1}{{\rm{k}}_2}} \right)^2}{\rm{ap}}\)

Thus it is a geometric progression.