Question

If $p^a=q^b=r^c=s^d$ and $p,q,r,s$ are in G.P. , then $a,b,c,d$ are in

• A. A.P
• B. G.P
• C. H.P
• D. None of the above

Answer 1

Answer: (C)

H.P

Since, $p,q,r,s$ are in G.P,

Let $p=m$, $q=mn$, $r=m{n}^2$ and $s=m{n}^3$.

Given that $p^a=q^b=r^c=s^d$,

$m^a={\left(mn\right)}^b={\left(m{n}^2\right)}^c={\left(m{n}^3\right)}^d$.

By taking logarithm with base $m$, we get

$a=b(1+{\log }_{m}n)=c(1+2{\log }_{m}n)=d(1+3{\log }_{m}n)$

Now, $a=b(1+{\log }_{m}n)$

$\Rightarrow {\log }_{m}n=\frac{a}{b}-1$

Substituting ${\log }_{m}n$ in $a=c(1+2{\log }_{m}n)$, we get

$\Rightarrow a=c(1+2\times \frac{a}{b}-2)$

$\Rightarrow b=\frac{2ac}{a+c}$

Therefore, $a,b,c$ are in H.P. Similarly, we can prove the same for $a,b,c,d$.
Hence, option C is correct.