If the roots of `10 x^3-n x^2-54 x-27=0` are in harmonic progression, then `n` eqauls _________.

Explanation

Given roots of the equation
`10x^(3)-cx^(2)-54x-27=0` are in HP ……..i
Now replaceing `x` by `1/x` in Eq I we get
`27x^(3)+54x^(2)+cx-10=0`…ii
Hence the roots of Eq. ii are in AP
Let `alha-d,a` and `a+d` are the roots of Eq ii
Then `a-d+a+a+d=-54/27` M
`impliesa=-2/3`........iii
Since a is a root of eq. ii then
`27a^(3)+54a^(2)+ca-10=0`
`implies27(-8/27)+54(4/9)+c(-2/3)-10=0`[ from Eq iii]
`implies6-(2c)/3=0` or `c=9`