n harmonic means have been inserted between 1 and 4 such that first meant:last mean=1:3,find n.

Answer 1

consider the n+2 terms of the corresponding arithmetic sequence with difference d.
d = (4-1)/(n+1) = 3/(n+1)
If we insert n harmonic means, then
1/H1 = 1 + 3/(n+1)
1/Hn = 1 + n*3/(n+1)
so H1/Hn = (1/(1 + 3/(n+1)))/(1/(1 + n*3/(n+1))) = 1/3
This gives n = 1/11 which is not an integer. Maybe you have your ratio backwards. The means between 1 and 1/4 will have decreasing values.

Answer 2

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√<span style="text-decoration:overline;"> X + 1 </span>
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Answer 3

Jee advance