Sequence And Series Question 433
Question: . An A.P. whose first term is unity and in which the sum of first half of any even number of terms to that of second half of the same number of terms is a constant ratio, then the common difference is:
Options:
A) 2
B) 1
C) 3
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Let $ S_{n} $ denote the sum of n terms of an A. P. According to given $ \frac{S_{n}}{S_{2n}-S_{n}}=k,\forall n\ge 1\Rightarrow \frac{S_1}{S_2-S_1}=\frac{S_2}{S_4-S_2} $
$ \Rightarrow ,S_1S_4-S_1S_2=S_2^{2}-S_1S_2\Rightarrow S_1S_4=S_2^{2} $
$ \Rightarrow a\frac{4}{2}[2a+(4-1)d]={{(a+a+d)}^{2}} $
$ \Rightarrow ,a(4a+6d)={{(2a+d)}^{2}} $
$ \Rightarrow 2ad=d^{2}\Rightarrow 2a=d $ Since $ a=1, $ we get $ d=2 $
BETA
