Question

If ${a^x} = {b^y} = {c^z}$ and a,b,care in G.P. then x,y,z are in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hint
Every word in a geometric progression or sequence is altered by another term by a common ratio. In our case, we are provided that ${a^x} = {b^y} = {c^z}$ and a,b,c are in Geometric progression. And we are to determine what progression is x,y,z. For that, we have to use the form that ${b^2} = ac$ and solve accordingly to get the desired answer.
Formula used:
As a,b,c are in the geometric progression we can use the formula
${b^2} = ac$
Complete step-by-step solution
The given series of progression is a,b,cand ${a^x} = {b^y} = {c^z}$
Let the series be
${a^x} = {b^y} = {c^z} = p$
$a=p^\left(\dfrac{1}{x}\right)$
$b=p^\left(\dfrac{1}{y}\right)$ and
$c=p^\left(\dfrac{1}{z}\right)$
As a,b,c are in the geometric progression
${b^2} = ac$
The equation becomes
$(p^{\frac{1}{y}})^{2}=(p^{\frac{1}{x}}.p^{\frac{1}{z}})$
$(p^{\frac{2}{y}})=(p^{\frac{1}{x}+\frac{1}{z}})$
P has the equating powers as
${\dfrac{2}{y}}={\dfrac{1}{x}+\dfrac{1}{z}}$
The H.P series are in terms x,y,z.
Therefore, the correct option is C
Note
Students are likely to make mistakes in these types of problems because it includes progression concepts. First the difficulty arises in finding which series is the given and what formula to use. so, one should be cautious in applying formulas. Here, we are given that the series in G.P. So, we can use the formula ${b^2} = ac$ _{to get the desired answer.} Also, it should be noted that a common ratio between consecutive terms in GP is used to identify the series.