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 \(=\cfrac{\frac{m(a+b)}2}{\frac{n(a+b)}2}\)
\(=\frac ma=m:n\)
