Let \[{\rm{a}},{\rm{b}},{\rm{c}}\] be in A.P. with a common difference d. Then \[{e^{1/c}},{e^{b/ac}},{e^{1/a}}\] are in
A. G.P. with common ratio \[{e^d}\]
B. G.P. with common ratio \[{e^{1/d}}\]
C. G.P. with common ratio \[{e^{d/\left( {{b^2} - {d^2}} \right)}}\]
D. A.P.

Hint: You must understand the concept of arithmetic progression as well as the initial term of AP, common difference of AP, and sum of n terms of AP in order to solve this issue. In our case, we are provided the series \[{e^{1/c}},{e^{b/ac}},{e^{1/a}}\] and are asked to determine the progression in possess. For that we have to use the general form \[2b = a + c\] as we are provided that \[{\rm{a}},{\rm{b}},{\rm{c}}\] are in A.P. with a common difference d to determine the desired solution.

Formula Used: The progression ca be determined using the formula
\[2b = a + c\]

Complete step by step solution: We have been provided in the question that,
Let \[{\rm{a}},{\rm{b}},{\rm{c}}\] be in A.P. with a common difference d
Now, we have already known that if \[{\rm{a}},{\rm{b}},{\rm{c}}\] are in A.P, then
\[2b = a + c\]
Now, we have to write the given terms in the form of \[2b = a + c\] we have
\[{e^{1/c}} \times {e^{1/a}} = {e^{(a + c)/ac}} = {e^{2b/ac}} = {\left( {{e^{b/ac}}} \right)^2}\]
Since, the terms \[{e^{1/c}},{e^{b/ac}},{e^{1/a}}\] are in Geometric progression with common ratio
\[ = \dfrac{{{e^{b/ac}}}}{{{e^{1/c}}}}\]
Now, we have to move the denominator to the numerator, we get
\[ = {e^{(b - a)/ac}}\]
On substituting the corresponding values, we get
\[ = {e^{d/\left( {{b^2} - {d^2}} \right)}}\]
Now, we have to expand the exponents in the above equation, we get
\[ = {e^{d/(b - d)(b + d)}}\]
Therefore, \[{e^{1/c}},{e^{b/ac}},{e^{1/a}}\] are in G.P. with common ratio \[{e^{d/\left( {{b^2} - {d^2}} \right)}}\]

Option ‘C’ is correct

Note: The arithmetic progression arrangement, which is based on the first term and common difference, is something we need to be aware of. Students are likely to make mistakes in these types of problems because it includes powers and exponents. That makes the calculation bit difficult to solve. So, one should be very cautious while solving problems include power calculations to get the desired answer.