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If the sum of n terms of sequence Sn = n...

If the sum of n terms of sequence `S_n = n/(n+1)`

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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.
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