Number of positive terms in the sequence `x_n=195/(4P_n)-(n+3p_3)/(P_(n+1)), n in N` (here `p_n=|anglen`)

Answer

4

Explanation

We have, `x_(n)=(195)/(4P_(n))-(.^(n+3)A_(3))/(P_(n+1))`
`thereforex_(n)=(195)/(4*n!)-((n+3)(n+2)(n+1))/((n+1)!)`
`=-(195)/(4*n!)-((n+3)(n+2))/(n!)`
`=(195-4n^(2)-20n-24)/(4*n!)=(171-4n^(2)-20n)/(4*n!)`
`becausex_(n)` is positive
`therefore(171-4n^(2)-20n)/(4*n!) gt0`
`implies4n^(2)+20n-171 lt0`
which is true for n=1,2,3,4
Hence, the given sequence `(x_(n))` has 4 positive terms.