Sequence And Series Question 642
Question: If $ x,\ y,\ z $ are in H.P., then the value of expression $ \log (x+z)+\log (x-2y+z) $ will be
[RPET 1985, 2000]
Options:
A) $ \log (x-z) $
B) $ 2\log (x-z) $
C) $ 3\log (x-z) $
D) $ 4\log (x-z) $
Show Answer
Answer:
Correct Answer: B
Solution:
If $ x,\ y,,z $ are in H.P., then $ y=\frac{2xz}{x+z} $ Now, $ {\log_{e}}(x+z)+{\log_{e}}(x-2y+z) $ $ \sum\limits_{i=1}^{n}{{}}=3\sum\limits_{i=1}^{n}{i}-2\sum\limits_{i=1}^{n}{1}=3\frac{n(n+1)}{2}-2n=\frac{n(3n-1)}{2} $ $ ={\log_{e}}[ (x+z),( x+z-\frac{4xz}{x+z} ) ] $ $ ={\log_{e}}[{{(x+z)}^{2}}-4xz]={\log_{e}}{{(x-z)}^{2}}=2{\log_{e}}(x-z) $ .
BETA
