Sequence And Series Question 590
Question: If the sum of an infinite G.P. and the sum of square of its terms is 3, then the common ratio of the first series is
[Roorkee 1972]
Options:
A) 1
B) $ \frac{1}{2} $
C) $ \frac{2}{3} $
D) $ \frac{3}{2} $
Show Answer
Answer:
Correct Answer: B
Solution:
Let the first series be $ a+ar+ar^{2}+……… $ then the second series is $ a^{2}+a^{2}r^{2}+a^{2}r^{4}+………. $ their sums are given as 3. So, we have $ \frac{a}{1-r}=3 $ or $ a=3(1-r) $ and $ \frac{q}{2}=\frac{pr}{p+r}=K $ or $ a^{2}=3(1-r^{2}) $ Eliminating $ a,\ {{{ 3,(1-r) }}^{2}}=3,(1-r^{2}) $
$ \Rightarrow $ $ 3,(1-r)=(1+r) $ , $ { \because \ r\ne 1 } $
$ \Rightarrow $ $ 4r=2 $ or $ r=\frac{1}{2} $ .
BETA
