Sequence And Series Question 526
Question: If $ A $ be a arithmetic mean between two numbers and $ S $ be the sum of $ n $ arithmetic means between the same numbers, then
Options:
A) $ S=n,A $
B) $ A=n,S $
C) $ A=S $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let the two quantities be $ a $ and $ b $ and let $ A_1,\ A_2…….,A_{n} $ be the $ n $ A.M.’s between them. Then $ a,\ A_1,\ A_2……A_{n},\ b $ are in A.P. and let $ d $ be the common difference. Now $ {T_{n+2}}=b=a+(n+2-1)d\Rightarrow d=\frac{b-a}{n+1} $ Also $ A_1+A_2+……+A_{n}={S_{n+1}}-a $ $ =\frac{1}{2}(n+1)[ 2a+(n+1-1)\frac{(b-a)}{(n+1)} ]-a $ = $ \frac{n}{2}[2a+(b-a)]=\frac{n}{2}(a+b)=n( \frac{a+b}{2} )=nA $ . Trick: Let 1, 3, 5, 7, 9 is in A.P. In this series $ A=5,n=3,S=15 $
Þ $ S=nA $ .
BETA
