Sum of 10 arithmetic means between 3, 10 is  A) 65 

Correct option is (A) 65

Let 10 arithmetic means between 3 & 10 are a+d, a+2d, a+3d, ........, a+10d.

\(\therefore\) 3, a+d, a+2d, a+3d, ........, a+10d, 10 will form an A.P.

\(\therefore a=3\;\&\;a+11d=10\)

\(\Rightarrow 11d=10-a\)

\(=10-3=7\)

\(\Rightarrow d=\frac7{11}\)

Now, sum of 10 arithmetic means

= (a+d) + (a+2d) + (a+3d) + ........ + (a+10d)

\(=10a+d\,(1+2+3+.......+10)\)

\(=10a+\frac{10\times11}2\,d\)          \((\because1+2+3+........+n=\frac{n(n+1)}2)\)

\(=10a+55d\)

\(=10\times3+55\times\frac7{11}\)          \((\because a=3\;\&\;d=\frac7{11})\)

\(=30+35=65\)

Hence, sum of 10 arithmetic means between 3 & 10 is 65.

Correct option is A) 65