Sequences & Series
Logarithmic AP — Finding Ratio a:b:c
nta_pyq_2024_jan
Grade 11
Question:
If $\log_e a,\log_e b,\log_e c$ are in an A.P. and $\log_e a-\log_e 2b,\log_e 2b-\log_e 3c,\log_e 3c-\log_e a$ are also in an A.P., then $a:b:c$ is equal to
9:6:4
16:4:1
25:10:4
6:3:2
Step-by-Step Solution
Key Concept: $\log a,\log b,\log c$ in AP $\Rightarrow b^2=ac$ ...(i). Second AP condition: $\left(\frac{2b}{3c}\right)^2=\frac{a}{2b}\cdot\frac{3c}{a}\Rightarrow\frac{b}{c}=\frac{3}{2}$...(ii). From (i) and (ii): $a/b=3/2$. So $a:b:c=9:6:4$.
$b/c=3/2$, $a/b=3/2$. $a:b:c=9:6:4$.
Correct Answer: 1