Sequences and Series 5 Question 4
4. If $x>1, y>1, z>1$ are in GP, then $\frac{1}{1+\ln x}, \frac{1}{1+\ln y}$, $\frac{1}{1+\ln z}$ are in
(1998, 2M)
(a) AP
(b) HP
(c) GP (d) None of these
Objective Question II
(One or more than one correct option)
Show Answer
Answer:
Correct Answer: 4. (b)
Solution:
- Let the common ratio of the GP be $r$. Then,
$$ y=x r \text { and } z=x r^{2} $$
$\Rightarrow \ln y=\ln x+\ln r$ and $\ln z=\ln x+2 \ln r$
Let
$$ A=1+\ln x, D=\ln r $$
Then, $\frac{1}{1+\ln x}=\frac{1}{A}, \frac{1}{1+\ln y}=\frac{1}{1+\ln x+\ln r}=\frac{1}{A+D}$
and $\quad \frac{1}{1+\ln z}=\frac{1}{1+\ln x+2 \ln r}=\frac{1}{A+2 D}$
Therefore, $\frac{1}{1+\ln x}, \frac{1}{1+\ln y}, \frac{1}{1+\ln z}$ are in HP.
