The largest term in the sequence `a_n=(n^2)/(n^3+200)` is given by `(529)/(49)` (b) `8/(89)` `(49)/(543)` (d) none of these

Options

A. `49/543`
B. `8/89`
C. `1/52`
D. non of these

Answer

A

Explanation

Let f (n)=`a_n=(n^2)/(n^3 + 200),n in N`.
`f(n)=(2(n^3+200)n-3n^4)/(n^3+200)=n(400-n^3)/(n^3+200)^2`

Clearly f'(n) `ne ` 0 for any n `in ` N .So f(n) has no extreme point
But
`f(n) gt 0 " for " n le 7` and decresing for `n le 8`
Now `f(7)=(49)/(543) and f(8)=(8)/(89)`
Clearly , `f(7) gt f(8)`
Hence , the value of the largest term is `49/543`