Question

Given \[{a^x} = {b^y} = {c^z} = {d^u}\] and \[a,b,c,d\] are in G.P., then \[x,y,z,u\] are in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hint:
We begin by remembering that a, b, c and d are in G.P (Geometric Progression), and assuming \[b = ar,c = a{r^2},d = a{r^3}\] and proceed by taking logarithms for the stated information to solve further. Ad we equate the logarithmic equation to k and then solve each term with respect to k to determine the required progression.
Formula used:
 if a, b, c and d are in G.P (Geometric Progression), and assuming
\[b = ar,c = a{r^2},d = a{r^3}\]
\[ \Rightarrow (\log ab) = (\log a + \log b) \]
Complete step-by-step solution:
To be able to answer this question, one must be familiar with concepts such as progressions. Take care of the calculations so that you are certain of the final result. We need to understand concepts of logarithms.
We have been given in the question that,
The sequence \[{a^x} = {b^y} = {c^z} = {d^u}\]and\[a,b,c,d\] are in Geometric progression.
Since,\[a,b,c,d\] are in Geometric progression.
Let us assume that,
 \[b = ar,c = a{r^2},d = a{r^3}\]
Taking logarithm for the equation\[b = {\rm{ ar }}\quad c = a{r^2}\quad d = a{r^3}\], we get
\[ \Rightarrow x(\log a) = y(\log a + \log r) = z(\log a + 2\log r) = u(\log a + 3\log r) = k\]--- (1)
Now, we have to solve each term with k from the above equation:
Therefore, for each term of the equation (1), we have to equate the sequence with \[k\]:
\[x = \frac{k}{{\log a}}; y = \frac{k}{{\log a + \log r}};z = \frac{k}{{\log a + 2\log r}}; u = \frac{k}{{\log a + 3\log r}}\]
Since, \[\log a,\log a + \log r,\log a + 2\log r,\log a + 3\log r\] are in Arithmetic progression
Therefore, \[{\rm{x}},{\rm{y}},{\rm{z}},{\rm{u}}\]are in Harmonic progression.
Hence, the option C is correct.


Note:
The problem can also be solved by assuming the common ratio of \[a\],\[b\] and \[c\] as \[r\]. To find the roots of the quadratic equation \[a{x^2} + 2bx + c = 0\], we make substitution \[b = ar,c = a{r^2}\]. We then substitute those roots in the quadratic equation and solve the problem again. We should not choose numbers at random for \[abc\] because this complicates the calculation.