Question

If p, q, r are in A.P., then pth,qth and rth terms of any G.P. are themselves in

Answer 1

Hint:To find the terms of a geometric progression, we need to use the formula of: \[T=a{{r}^{n-1}}\]

where \[a\] is the first element, \[r\] is the difference between the elements and \[n\] is the position of the element. After finding the terms we will use the mid term method where we find the value of the middle term of three values which is square and equal to the product of the first and last term.

Complete step by step solution:
The values or terms are in A.P. hence, in A.P. now to find the values of the p th , q th and r th term we use the formula of \[T=a{{r}^{n-1}}\] as the terms are in G.P. Hence, the value of:

p th term is given as \[T=a{{r}^{n-1}}\]
\[\Rightarrow {{T}_{p}}=a{{r}^{p-1}}\]

q th term is given as \[T=a{{r}^{n-1}}\]
\[\Rightarrow {{T}_{q}}=a{{r}^{q-1}}\]

r th term is given as \[T=a{{r}^{n-1}}\]
\[\Rightarrow {{T}_{r}}=a{{r}^{r-1}}\]

Now after finding the values of each term, we find how the terms of G.P. are equated with each other by putting them in a equation of:

\[{{T}_{q}}^{2}={{T}_{p}}{{T}_{r}}\]

Placing the values in the above formula, we get the value of \[q\] with respect to \[p,r\] as:
\[\Rightarrow {{\left[ a{{r}^{q-1}} \right]}^{2}}=a{{r}^{p-1}}\left[ a{{r}^{r-1}} \right]\]
\[\Rightarrow \left[ {{r}^{q-1}} \right]\left[ {{r}^{q-1}} \right]={{r}^{p-1}}\left[ {{r}^{r-1}} \right]\]
\[\Rightarrow \left[ {{r}^{2\left( q-1 \right)}} \right]={{r}^{r+p-2}}\]

Now to prove that the terms \[p,q\] and \[r\] are in equation are given according to their power put together as:
\[\Rightarrow 2\left( q-1 \right)=r+p-2\]
\[\Rightarrow 2q=r+p\]

Hence, the above equation proves that \[p,q\] and \[r\] are in G.P.

Note:The geometric term are formed by multiplication of first and the difference whereas in A.P. the terms are formed by addition of first and the last term and when the sequence is arithmetic the terms are in successive addition whereas in geometric progression the terms are formed either by multiplication or by division.