The ratio of the sums of m arithmatic means and n arithmetic means between two numbers is
The ratio of the sums of m arithmatic means and n arithmetic means between two numbers is
A) n : m
B) 2m – 1 : 2n – 1
C) m : n
D) 2m + 1, 2n + 1
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Correct option is (C) m : n
Sum of m arithmatic means between a and b
\(=(a+d)+(a+2d)+.......+(a+md)\)
\(=ma+d\,(1+2+.....+m)\)
\(=ma+d\,\frac{m(m+1)}2\)
\(=ma+\frac{m(m+1)}2\times\frac{b-a}{m+1}\) \((\because d=\frac{b-a}{m+1})\)
\(=ma+\frac{m}2\,(b-a)\)
\(=\frac{2am+mb-ma}2\)
\(=\frac{m(a+b)}2\)
\(\therefore\) Sum of n arithmatic means between a and b \(=\frac{n(a+b)}2\)
\(\therefore\) Required ratio \(=\cfrac{\frac{m(a+b)}2}{\frac{n(a+b)}2}\)
\(=\frac ma=m:n\)
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