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Equal mth Means Between a,2b and 2a,b: Find a:b | JEE

JEE Maths question with a full step-by-step solution.

Question
If nn arithmetic means are inserted between aa and 2b2b, and also between 2a2a and bb (a,b∈Ra,b\in\mathbb{R}), and the mmth means of the two sets are equal, then a:ba:b equals
An−m+1:mn-m+1:m
Bn−m+1:nn-m+1:n
Cm:n−m+1m:n-m+1correct
Dn:n−m+1n:n-m+1
Solution
Step 1: The mmth means of the two sets are
Am=a+m⋅2b−an+1,Bm=2a+m⋅b−2an+1.A_m=a+m\cdot\frac{2b-a}{n+1},\qquad B_m=2a+m\cdot\frac{b-2a}{n+1}.
 Step 2: Set Am=BmA_m=B_m and clear (n+1)(n+1):
a(n+1)+m(2b−a)=2a(n+1)+m(b−2a) ⇒ m(a+b)=a(n+1).a(n+1)+m(2b-a)=2a(n+1)+m(b-2a)\ \Rightarrow\ m(a+b)=a(n+1).
 Step 3:
mb=a(n+1−m) ⇒ ab=mn−m+1 ⇒ a:b=m:(n−m+1).mb=a(n+1-m)\ \Rightarrow\ \frac{a}{b}=\frac{m}{n-m+1}\ \Rightarrow\ a:b=m:(n-m+1).
 Correct answer: (3)
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